Preliminaries About Functions

Noel DeJarnette and Austin Rochford

August 21, 2012 Navigation Symbols

∈, ”in” or ”an element of”, 1 {0, 1, 2, 3, 4} ⊂, ”subset of” or ”contained in”, {0, 1, 2, 3, 4} ⊂ N ⊂ Z ⊂ Q ⊂ R (a, b), interval notation for {x ∈ R : a < x < b},(−∞, ∞) is all of R. {x ∈ R : x ≥ a}, set notation for [a, ∞). N, natural numbers, {1, 2, 3, 4,...} Z, integers, {..., −4, −3, −2, −1, 0, 1, 2, 3, 4,...} n p o Q, rational numbers, x ∈ R : x = q where p, q ∈ Z, q 6= 0 so f takes pts from a set, A, of the real numbers (it could be all of R

Functions, a Doodle

The functions we will deal with in Calc 1 (and probably all the functions you have seen and will see until Calc 3) send one real number to another real number. so f takes pts from a set, A, of the real numbers (it could be all of R

Functions, a Doodle

The functions we will deal with in Calc 1 (and probably all the functions you have seen and will see until Calc 3) send one real number to another real number. Functions, a Doodle

The functions we will deal with in Calc 1 (and probably all the functions you have seen and will see until Calc 3) send one real number to another real number.

so f takes pts from a set, A, of the real numbers (it could be all of R Functions, a Doodle

The functions we will deal with in Calc 1 (and probably all the functions you have seen and will see until Calc 3) send one real number to another real number.

so f takes pts from a set, A, of the real numbers (it could be all of R) to another set, B, also in the reals.

Functions, a Doodle

The functions we will deal with in Calc 1 (and probably all the functions you have seen and will see until Calc 3) send one real number to another real number.

so f takes pts from a set, A, of the real numbers (it could be all of R) Functions, a Doodle

The functions we will deal with in Calc 1 (and probably all the functions you have seen and will see until Calc 3) send one real number to another real number.

so f takes pts from a set, A, of the real numbers (it could be all of R) to another set, B, also in the reals. Functions, a Doodle

The functions we will deal with in Calc 1 (and probably all the functions you have seen and will see until Calc 3) send one real number to another real number.

so f takes pts from a set, A, of the real numbers (it could be all of R) to another set, B, also in the reals. Functions, a Doodle

The functions we will deal with in Calc 1 (and probably all the functions you have seen and will see until Calc 3) send one real number to another real number.

More precisely, a number a ∈ A Functions, a Doodle

The functions we will deal with in Calc 1 (and probably all the functions you have seen and will see until Calc 3) send one real number to another real number.

More precisely, a number a ∈ A Functions, a Doodle

The functions we will deal with in Calc 1 (and probably all the functions you have seen and will see until Calc 3) send one real number to another real number.

More precisely, a number a ∈ A is sent to a point f (a) = b ∈ B Functions, a Doodle

The functions we will deal with in Calc 1 (and probably all the functions you have seen and will see until Calc 3) send one real number to another real number.

More precisely, a number a ∈ A is sent to a point f (a) = b ∈ B (meaning f (x) exists)

This would be the set A in the doodle. Example The function f (x) = x2 is defined everywhere, so the domain is all of R.

Example f (x) = ln x is defined only for strictly positive numbers so the domain is {x ∈ R : x > 0}.

Domain and Range of Functions

Definition The domain of a function f is the set of points where f is defined, . This would be the set A in the doodle. Example The function f (x) = x2 is defined everywhere, so the domain is all of R.

Example f (x) = ln x is defined only for strictly positive numbers so the domain is {x ∈ R : x > 0}.

Domain and Range of Functions

Definition The domain of a function f is the set of points where f is defined,(meaning f (x) exists). Example The function f (x) = x2 is defined everywhere, so the domain is all of R.

Example f (x) = ln x is defined only for strictly positive numbers so the domain is {x ∈ R : x > 0}.

Domain and Range of Functions

Definition The domain of a function f is the set of points where f is defined,(meaning f (x) exists).

This would be the set A in the doodle. Example f (x) = ln x is defined only for strictly positive numbers so the domain is {x ∈ R : x > 0}.

Domain and Range of Functions

Definition The domain of a function f is the set of points where f is defined,(meaning f (x) exists).

This would be the set A in the doodle. Example The function f (x) = x2 is defined everywhere, so the domain is all of R. Domain and Range of Functions

Definition The domain of a function f is the set of points where f is defined,(meaning f (x) exists).

This would be the set A in the doodle. Example The function f (x) = x2 is defined everywhere, so the domain is all of R.

Example f (x) = ln x is defined only for strictly positive numbers so the domain is {x ∈ R : x > 0}. This would be the set B in the doodle. Example The function f (x) = x2 is always nonnegative, so the range is {x ∈ R : x ≥ 0}.

Question What does nonnegative mean? How does the domain of ln x differ from the range of x2?

Example The function f (x) = ln x gives all values from −∞ to ∞ so the range is all of R.

Domain and Range of Functions

Definition The range of a function f is the set of points that are the values f (x). Example The function f (x) = x2 is always nonnegative, so the range is {x ∈ R : x ≥ 0}.

Question What does nonnegative mean? How does the domain of ln x differ from the range of x2?

Example The function f (x) = ln x gives all values from −∞ to ∞ so the range is all of R.

Domain and Range of Functions

Definition The range of a function f is the set of points that are the values f (x).

This would be the set B in the doodle. Question What does nonnegative mean? How does the domain of ln x differ from the range of x2?

Example The function f (x) = ln x gives all values from −∞ to ∞ so the range is all of R.

Domain and Range of Functions

Definition The range of a function f is the set of points that are the values f (x).

This would be the set B in the doodle. Example The function f (x) = x2 is always nonnegative, so the range is {x ∈ R : x ≥ 0}. Example The function f (x) = ln x gives all values from −∞ to ∞ so the range is all of R.

Domain and Range of Functions

Definition The range of a function f is the set of points that are the values f (x).

This would be the set B in the doodle. Example The function f (x) = x2 is always nonnegative, so the range is {x ∈ R : x ≥ 0}.

Question What does nonnegative mean? How does the domain of ln x differ from the range of x2? Example The function f (x) = ln x gives all values from −∞ to ∞ so the range is all of R.

Domain and Range of Functions

Definition The range of a function f is the set of points that are the values f (x).

This would be the set B in the doodle. Example The function f (x) = x2 is always nonnegative, so the range is {x ∈ R : x ≥ 0}.

Question What does nonnegative mean? How does the domain of ln x differ from the range of x2? Domain and Range of Functions

Definition The range of a function f is the set of points that are the values f (x).

This would be the set B in the doodle. Example The function f (x) = x2 is always nonnegative, so the range is {x ∈ R : x ≥ 0}.

Question What does nonnegative mean? How does the domain of ln x differ from the range of x2?

Example The function f (x) = ln x gives all values from −∞ to ∞ so the range is all of R. The Preimage is a perfect example. It will be used to find inverses of functions and is important in finding the domains of compositions.

Domain and Range of Functions

Sometimes we want information about subsets of the range or the domain. Domain and Range of Functions

Sometimes we want information about subsets of the range or the domain. The Preimage is a perfect example. It will be used to find inverses of functions and is important in finding the domains of compositions. Domain and Range of Functions

Sometimes we want information about subsets of the range or the domain. The Preimage is a perfect example. It will be used to find inverses of functions and is important in finding the domains of compositions. First, look at subset of the range, C.

Domain and Range of Functions

The Preimage will be a subset of the domain. Domain and Range of Functions

The Preimage will be a subset of the domain. First, look at subset of the range, C. Domain and Range of Functions

The Preimage will be a subset of the domain. First, look at subset of the range, C. Domain and Range of Functions

The Preimage will be a subset of the domain. First, look at subset of the range, C. The Preimage of C is all the points in A that f sends to C. If all the points of D are sent to C, and those are the only points sent to C, then D is the preimage of C

Domain and Range of Functions

The Preimage will be a subset of the domain. First, look at subset of the range, C. The Preimage of C is all the points in A that f sends to C. Domain and Range of Functions

The Preimage will be a subset of the domain. First, look at subset of the range, C. The Preimage of C is all the points in A that f sends to C.

If all the points of D are sent to C, and those are the only points sent to C, then D is the preimage of C Domain and Range of Functions

The Preimage will be a subset of the domain. First, look at subset of the range, C.The Preimage of C is all the points in A that f sends to C.

Formally, Preimage(C) = {x ∈ A : f (x) ∈ C}. You aren’t able to deduce much about a function if all you know is that it sends (0, ∞) (0, ∞). This is where the graph of the function comes in.

Graphs of Functions

This picture might be helpful in understanding what functions do, but it doesn’t really explain how the function behaves. This is where the graph of the function comes in.

Graphs of Functions

This picture might be helpful in understanding what functions do, but it doesn’t really explain how the function behaves. You aren’t able to deduce much about a function if all you know is that it sends (0, ∞) (0, ∞). Graphs of Functions

This picture might be helpful in understanding what functions do, but it doesn’t really explain how the function behaves. You aren’t able to deduce much about a function if all you know is that it sends (0, ∞) (0, ∞). This is where the graph of the function comes in. Graphs of Functions

This picture might be helpful in understanding what functions do, but it doesn’t really explain how the function behaves. You aren’t able to deduce much about a function if all you know is that it sends (0, ∞) (0, ∞). This is where the graph of the function comes in. Graphs of Functions

This picture might be helpful in understanding what functions do, but it doesn’t really explain how the function behaves. You aren’t able to deduce much about a function if all you know is that it sends (0, ∞) (0, ∞). This is where the graph of the function comes in. Graphs of Functions

This picture might be helpful in understanding what functions do, but it doesn’t really explain how the function behaves. You aren’t able to deduce much about a function if all you know is that it sends (0, ∞) (0, ∞). This is where the graph of the function comes in. Graphs of Functions

This picture might be helpful in understanding what functions do, but it doesn’t really explain how the function behaves. We aren’t able to deduce much about a function if all we know is that it sends (0, ∞) (0, ∞). This is where the graph of the function comes in. The points in the set are the pair (a, f (a), a point in the domain with its associated point in the range.

Graphs of Functions

2 The graph of a function is a set in the cartesian plane (R ) Graphs of Functions

2 The graph of a function is a set in the cartesian plane (R ) The points in the set are the pair (a, f (a), a point in the domain with its associated point in the range. Graphs of Functions

2 The graph of a function is a set in the cartesian plane (R ) The points in the set are the pair (a, f (a), a point in the domain with its associated point in the range. Graphs of Functions

2 The graph of a function is a set in the cartesian plane (R ) The points in the set are the pair (a, f (a), a point in the domain with its associated point in the range. Plot all the points and we have an informative way to visually describe a function. Graphs of Functions

2 The graph of a function is a set in the cartesian plane (R ) The points in the set are the pair (a, f (a), a point in the domain with its associated point in the range. Plot all the points and we have an informative way to visually describe a function. If the x-axis represents the inputs and the y-axis represents the values of f (x), or the outputs We can easily see that the range is all values greater than or equal to 0. It is harder to see the domain, but since we don’t see any discontinuities, we assume it keeps extending, and the domain is all real numbers.

Deducing Domains by Inspection

Let’s begin with examples we have already seen. If the x-axis represents the inputs and the y-axis represents the values of f (x), or the outputs We can easily see that the range is all values greater than or equal to 0. It is harder to see the domain, but since we don’t see any discontinuities, we assume it keeps extending, and the domain is all real numbers.

Deducing Domains by Inspection

Let’s begin with examples we have already seen. We can easily see that the range is all values greater than or equal to 0. It is harder to see the domain, but since we don’t see any discontinuities, we assume it keeps extending, and the domain is all real numbers.

Deducing Domains by Inspection

Let’s begin with examples we have already seen.

If the x-axis represents the inputs and the y-axis represents the values of f (x), or the outputs It is harder to see the domain, but since we don’t see any discontinuities, we assume it keeps extending, and the domain is all real numbers.

Deducing Domains by Inspection

Let’s begin with examples we have already seen.

If the x-axis represents the inputs and the y-axis represents the values of f (x), or the outputs We can easily see that the range is all values greater than or equal to 0. Deducing Domains by Inspection

Let’s begin with examples we have already seen.

If the x-axis represents the inputs and the y-axis represents the values of f (x), or the outputs We can easily see that the range is all values greater than or equal to 0. It is harder to see the domain, but since we don’t see any discontinuities, we assume it keeps extending, and the domain is all real numbers. We can easily same for f (x) = −x − 1

In fact all polynomials will have a domain of all real numbers. Question What polynomials will have a range of all real numbers?

Aside about Polynomials

2 We just saw that the domain of f (x) = x is R. In fact all polynomials will have a domain of all real numbers. Question What polynomials will have a range of all real numbers?

Aside about Polynomials

2 We just saw that the domain of f (x) = x is R. We can easily same for f (x) = −x − 1 In fact all polynomials will have a domain of all real numbers. Question What polynomials will have a range of all real numbers?

Aside about Polynomials

2 We just saw that the domain of f (x) = x is R. We can easily same for f (x) = −x − 1 Question What polynomials will have a range of all real numbers?

Aside about Polynomials

2 We just saw that the domain of f (x) = x is R. We can easily same for f (x) = −x − 1

In fact all polynomials will have a domain of all real numbers. Aside about Polynomials

2 We just saw that the domain of f (x) = x is R. We can easily same for f (x) = −x − 1

In fact all polynomials will have a domain of all real numbers. Question What polynomials will have a range of all real numbers? We can see there are no values for the function to the left of zero. The domain is {x ∈ R : x ≥ 0}. Same for the range. Question Can you find a function whose domain is not all real numbers, but whose range is?

Deducing Domains by Inspection

Let’s look at a function whose domain is not all of R. We can see there are no values for the function to the left of zero. The domain is {x ∈ R : x ≥ 0}. Same for the range. Question Can you find a function whose domain is not all real numbers, but whose range is?

Deducing Domains by Inspection

Let’s look at a function whose domain is not all of R. Same for the range. Question Can you find a function whose domain is not all real numbers, but whose range is?

Deducing Domains by Inspection

Let’s look at a function whose domain is not all of R.

We can see there are no values for the function to the left of zero. The domain is {x ∈ R : x ≥ 0}. Question Can you find a function whose domain is not all real numbers, but whose range is?

Deducing Domains by Inspection

Let’s look at a function whose domain is not all of R.

We can see there are no values for the function to the left of zero. The domain is {x ∈ R : x ≥ 0}. Same for the range. Deducing Domains by Inspection

Let’s look at a function whose domain is not all of R.

We can see there are no values for the function to the left of zero. The domain is {x ∈ R : x ≥ 0}. Same for the range. Question Can you find a function whose domain is not all real numbers, but whose range is? After all, the outputs of functions we have seen so far are just elements in the real numbers. Question What is the domain of h(x) = f (x) + g(x) in terms of the domains of f and g?

Both f (x) AND g(x) need to be defined for h(x) to exist. Since both need to be defined we have the domain of h(x) = domain f T domain g. Question f (x) How would you have to adjust the domain for h(x) = g(x) ?

Domains of Combined Functions

We can add, subtract, multiply, and divide functions just as we can numbers. Question What is the domain of h(x) = f (x) + g(x) in terms of the domains of f and g?

Both f (x) AND g(x) need to be defined for h(x) to exist. Since both need to be defined we have the domain of h(x) = domain f T domain g. Question f (x) How would you have to adjust the domain for h(x) = g(x) ?

Domains of Combined Functions

We can add, subtract, multiply, and divide functions just as we can numbers. After all, the outputs of functions we have seen so far are just elements in the real numbers. Both f (x) AND g(x) need to be defined for h(x) to exist. Since both need to be defined we have the domain of h(x) = domain f T domain g. Question f (x) How would you have to adjust the domain for h(x) = g(x) ?

Domains of Combined Functions

We can add, subtract, multiply, and divide functions just as we can numbers. After all, the outputs of functions we have seen so far are just elements in the real numbers. Question What is the domain of h(x) = f (x) + g(x) in terms of the domains of f and g? Since both need to be defined we have the domain of h(x) = domain f T domain g. Question f (x) How would you have to adjust the domain for h(x) = g(x) ?

Domains of Combined Functions

We can add, subtract, multiply, and divide functions just as we can numbers. After all, the outputs of functions we have seen so far are just elements in the real numbers. Question What is the domain of h(x) = f (x) + g(x) in terms of the domains of f and g?

Both f (x) AND g(x) need to be defined for h(x) to exist. Question f (x) How would you have to adjust the domain for h(x) = g(x) ?

Domains of Combined Functions

We can add, subtract, multiply, and divide functions just as we can numbers. After all, the outputs of functions we have seen so far are just elements in the real numbers. Question What is the domain of h(x) = f (x) + g(x) in terms of the domains of f and g?

Both f (x) AND g(x) need to be defined for h(x) to exist. Since both need to be defined we have the domain of h(x) = domain f T domain g. Domains of Combined Functions

We can add, subtract, multiply, and divide functions just as we can numbers. After all, the outputs of functions we have seen so far are just elements in the real numbers. Question What is the domain of h(x) = f (x) + g(x) in terms of the domains of f and g?

Both f (x) AND g(x) need to be defined for h(x) to exist. Since both need to be defined we have the domain of h(x) = domain f T domain g. Question f (x) How would you have to adjust the domain for h(x) = g(x) ? This is most helpful when exploring translations and reflections of graphs of functions, but it will come up in many other contexts. Example Let f (x) = ln x and g(x) = x2, then h(x) = g ◦ f (x) = g(f (x)) = g(ln x) = (ln x)2 = ln2 x.

Question What conditions must be met for h(x) to exist? (i.e. what is the domain of h(x))

Domains of Compositions of Functions

Compositions of functions is simply where we use one function as the input for another function. Example Let f (x) = ln x and g(x) = x2, then h(x) = g ◦ f (x) = g(f (x)) = g(ln x) = (ln x)2 = ln2 x.

Question What conditions must be met for h(x) to exist? (i.e. what is the domain of h(x))

Domains of Compositions of Functions

Compositions of functions is simply where we use one function as the input for another function. This is most helpful when exploring translations and reflections of graphs of functions, but it will come up in many other contexts. Question What conditions must be met for h(x) to exist? (i.e. what is the domain of h(x))

Domains of Compositions of Functions

Compositions of functions is simply where we use one function as the input for another function. This is most helpful when exploring translations and reflections of graphs of functions, but it will come up in many other contexts. Example Let f (x) = ln x and g(x) = x2, then h(x) = g ◦ f (x) = g(f (x)) = g(ln x) = (ln x)2 = ln2 x. Domains of Compositions of Functions

Compositions of functions is simply where we use one function as the input for another function. This is most helpful when exploring translations and reflections of graphs of functions, but it will come up in many other contexts. Example Let f (x) = ln x and g(x) = x2, then h(x) = g ◦ f (x) = g(f (x)) = g(ln x) = (ln x)2 = ln2 x.

Question What conditions must be met for h(x) to exist? (i.e. what is the domain of h(x)) Since the domain is all real numbers, that will clearly encompass the range of f . We conclude that the domain of the composition, h, is the same as the domain of f ,{x ∈ R : x > 0} Question Which statement is always true for any f and g: domain of f ⊂ domain of g ◦ f , domain of g ◦ f ⊂ domain of f ?

Question What is the domain of f ◦ g(x) = ln x2?

Domains of Compositions of Functions

g(x) = x2 is a polynomial, so its domain is all real numbers. We conclude that the domain of the composition, h, is the same as the domain of f ,{x ∈ R : x > 0} Question Which statement is always true for any f and g: domain of f ⊂ domain of g ◦ f , domain of g ◦ f ⊂ domain of f ?

Question What is the domain of f ◦ g(x) = ln x2?

Domains of Compositions of Functions

g(x) = x2 is a polynomial, so its domain is all real numbers. Since the domain is all real numbers, that will clearly encompass the range of f . Question Which statement is always true for any f and g: domain of f ⊂ domain of g ◦ f , domain of g ◦ f ⊂ domain of f ?

Question What is the domain of f ◦ g(x) = ln x2?

Domains of Compositions of Functions

g(x) = x2 is a polynomial, so its domain is all real numbers. Since the domain is all real numbers, that will clearly encompass the range of f . We conclude that the domain of the composition, h, is the same as the domain of f ,{x ∈ R : x > 0} Question What is the domain of f ◦ g(x) = ln x2?

Domains of Compositions of Functions

g(x) = x2 is a polynomial, so its domain is all real numbers. Since the domain is all real numbers, that will clearly encompass the range of f . We conclude that the domain of the composition, h, is the same as the domain of f ,{x ∈ R : x > 0} Question Which statement is always true for any f and g: domain of f ⊂ domain of g ◦ f , domain of g ◦ f ⊂ domain of f ? Domains of Compositions of Functions

g(x) = x2 is a polynomial, so its domain is all real numbers. Since the domain is all real numbers, that will clearly encompass the range of f . We conclude that the domain of the composition, h, is the same as the domain of f ,{x ∈ R : x > 0} Question Which statement is always true for any f and g: domain of f ⊂ domain of g ◦ f , domain of g ◦ f ⊂ domain of f ?

Question What is the domain of f ◦ g(x) = ln x2? Thus f (g(x)) will defined for all x such that g(x) > 0. The range of g is always NONNEGATIVE, which means we have to only worry about the x’s where g(x) = 0. So the domain of f ◦ g is {x ∈ R : x 6= 0} We see that domain f ◦ g ⊂ domain g

Domains of Compositions of Functions

f (x) = ln x has a domain of {x ∈ R : x > 0}. The range of g is always NONNEGATIVE, which means we have to only worry about the x’s where g(x) = 0. So the domain of f ◦ g is {x ∈ R : x 6= 0} We see that domain f ◦ g ⊂ domain g

Domains of Compositions of Functions

f (x) = ln x has a domain of {x ∈ R : x > 0}. Thus f (g(x)) will defined for all x such that g(x) > 0. So the domain of f ◦ g is {x ∈ R : x 6= 0} We see that domain f ◦ g ⊂ domain g

Domains of Compositions of Functions

f (x) = ln x has a domain of {x ∈ R : x > 0}. Thus f (g(x)) will defined for all x such that g(x) > 0. The range of g is always NONNEGATIVE, which means we have to only worry about the x’s where g(x) = 0. We see that domain f ◦ g ⊂ domain g

Domains of Compositions of Functions

f (x) = ln x has a domain of {x ∈ R : x > 0}. Thus f (g(x)) will defined for all x such that g(x) > 0. The range of g is always NONNEGATIVE, which means we have to only worry about the x’s where g(x) = 0. So the domain of f ◦ g is {x ∈ R : x 6= 0} Domains of Compositions of Functions

f (x) = ln x has a domain of {x ∈ R : x > 0}. Thus f (g(x)) will defined for all x such that g(x) > 0. The range of g is always NONNEGATIVE, which means we have to only worry about the x’s where g(x) = 0. So the domain of f ◦ g is {x ∈ R : x 6= 0} We see that domain f ◦ g ⊂ domain g √ √ We will see that x2 and x2 will have very different domains.

Domains of Compositions of Functions Graphically

We haven’t yet talked about inverses, but we should know that x2 √ √ √ and x are inverses, at least philosophically, so x2 = x2 where they are both defined. √ √ We will see that x2 and x2 will have very different domains.

Domains of Compositions of Functions Graphically

We haven’t yet talked about inverses, but we should know that x2 √ √ √ and x are inverses, at least philosophically, so x2 = x2 where they are both defined. Domains of Compositions of Functions Graphically

We haven’t yet talked about inverses, but we should know that x2 √ √ √ and x are inverses, at least philosophically, so x2 = x2 where they are both defined.

√ √ We will see that x2 and x2 will have very different domains. √ The Domain of x 2 √ Let’s first look at x2 which you might recognize as the absolute value, |x|. √ The Domain of x 2

The range of x2 are the nonnegative real numbers. √ The Domain of x 2

The range of x2 are the nonnegative real numbers. √ The Domain of x 2

√ Which coincides with the domain of x. √ The Domain of x 2

√ Which coincides with the domain of x. √ The Domain of x 2

√ Thus x2 is defined everywhere, (The domain is all real numbers). √ The Domain of x 2

What happens if we compose in the other order? √ The Domain of x 2

√ Right away we see that the domain of x2 will be restricted as the √ domain of x is x ≥ 0. √ The Domain of x 2

√ Right away we see that the domain of x2 will be restricted as the √ domain of x is x ≥ 0. √ The Domain of x 2

√ Right away we see that the domain of x2 will be restricted as the √ domain of x is x ≥ 0. √ The Domain of x 2

Again, the domain of x2 is all real numbers which contains the √ range of x. √ The Domain of x 2

Again, the domain of x2 is all real numbers which contains the √ range of x. √ The Domain of x 2

√ 2 So we conclude√ that the domain of x = {x ∈ R : x > 0}= 6 domain of x2 We will then use translations to generate a much larger class of graphs. If we know what the graph of f (x) looks like, then we will be able to find the graphs of −f (x), f (−x), f (x + c), f (x) + c where c is any constant. We would like to be able to do this by just transforming the graph and not computing values for the new function.

Translations via Compositions

It is important that we have the graphs, or at least rough sketches, of basic functions memorized. If we know what the graph of f (x) looks like, then we will be able to find the graphs of −f (x), f (−x), f (x + c), f (x) + c where c is any constant. We would like to be able to do this by just transforming the graph and not computing values for the new function.

Translations via Compositions

It is important that we have the graphs, or at least rough sketches, of basic functions memorized. We will then use translations to generate a much larger class of graphs. We would like to be able to do this by just transforming the graph and not computing values for the new function.

Translations via Compositions

It is important that we have the graphs, or at least rough sketches, of basic functions memorized. We will then use translations to generate a much larger class of graphs. If we know what the graph of f (x) looks like, then we will be able to find the graphs of −f (x), f (−x), f (x + c), f (x) + c where c is any constant. Translations via Compositions

It is important that we have the graphs, or at least rough sketches, of basic functions memorized. We will then use translations to generate a much larger class of graphs. If we know what the graph of f (x) looks like, then we will be able to find the graphs of −f (x), f (−x), f (x + c), f (x) + c where c is any constant. We would like to be able to do this by just transforming the graph and not computing values for the new function. We will see that f ◦ g = f (x + c) and g ◦ f = f (x) + c will be different functions. Horizontal, or left/right translations are formed by precomposing f with g, or f (x + c). If c is positive, then the graph of f will be shifted to the left. If c is negative, then the graph of f will be shifted to the right. Let’s look at an example.

Left and Right Translations

Let f (x) be the function we want to transform and g(x) = x + c. Horizontal, or left/right translations are formed by precomposing f with g, or f (x + c). If c is positive, then the graph of f will be shifted to the left. If c is negative, then the graph of f will be shifted to the right. Let’s look at an example.

Left and Right Translations

Let f (x) be the function we want to transform and g(x) = x + c. We will see that f ◦ g = f (x + c) and g ◦ f = f (x) + c will be different functions. If c is positive, then the graph of f will be shifted to the left. If c is negative, then the graph of f will be shifted to the right. Let’s look at an example.

Left and Right Translations

Let f (x) be the function we want to transform and g(x) = x + c. We will see that f ◦ g = f (x + c) and g ◦ f = f (x) + c will be different functions. Horizontal, or left/right translations are formed by precomposing f with g, or f (x + c). If c is negative, then the graph of f will be shifted to the right. Let’s look at an example.

Left and Right Translations

Let f (x) be the function we want to transform and g(x) = x + c. We will see that f ◦ g = f (x + c) and g ◦ f = f (x) + c will be different functions. Horizontal, or left/right translations are formed by precomposing f with g, or f (x + c). If c is positive, then the graph of f will be shifted to the left. Let’s look at an example.

Left and Right Translations

Let f (x) be the function we want to transform and g(x) = x + c. We will see that f ◦ g = f (x + c) and g ◦ f = f (x) + c will be different functions. Horizontal, or left/right translations are formed by precomposing f with g, or f (x + c). If c is positive, then the graph of f will be shifted to the left. If c is negative, then the graph of f will be shifted to the right. Left and Right Translations

Let f (x) be the function we want to transform and g(x) = x + c. We will see that f ◦ g = f (x + c) and g ◦ f = f (x) + c will be different functions. Horizontal, or left/right translations are formed by precomposing f with g, or f (x + c). If c is positive, then the graph of f will be shifted to the left. If c is negative, then the graph of f will be shifted to the right. Let’s look at an example. The graph of h(x) = (x + 2)2 will be shifted left two units.

Horizontal Translations, Example 1 (x 2)

This is the graph of f (x) = x2 Horizontal Translations, Example 1 (x 2)

This is the graph of f (x) = x2

The graph of h(x) = (x + 2)2 will be shifted left two units. h(−2) = f (0)

Horizontal Translations, Example 1 (x 2)

This is the graph of f (x) = x2

The graph of h(x) = (x + 2)2 will be shifted left two units. Horizontal Translations, Example 1 (x 2)

This is the graph of f (x) = x2

The graph of h(x) = (x + 2)2 will be shifted left two units. h(−2) = f (0) Horizontal Translations, Example 1 (x 2)

This is the graph of f (x) = x2

The graph of j(x) = (x − 3)2 will be shifted right three units. j(3) = f (0)

Horizontal Translations, Example 1 (x 2)

This is the graph of f (x) = x2

The graph of j(x) = (x − 3)2 will be shifted right three units. Horizontal Translations, Example 1 (x 2)

This is the graph of f (x) = x2

The graph of j(x) = (x − 3)2 will be shifted right three units. j(3) = f (0) The graph of h(x) = f (x + 2) will be shifted left two units.

Horizontal Translations, Example 2

This is the graph of some function f Horizontal Translations, Example 2

This is the graph of some function f

The graph of h(x) = f (x + 2) will be shifted left two units. h(−2) = f (0)

Horizontal Translations, Example 2

This is the graph of some function f

The graph of h(x) = f (x + 2) will be shifted left two units. Horizontal Translations, Example 2

This is the graph of some function f

The graph of h(x) = f (x + 2) will be shifted left two units. h(−2) = f (0) Horizontal Translations, Example 2

This is the graph of some function f

The graph of j(x) = f (x − 2) will be shifted right two units. j(2) = f (0)

Horizontal Translations, Example 2

This is the graph of some function f

The graph of j(x) = f (x − 2) will be shifted right two units. Horizontal Translations, Example 2

This is the graph of some function f

The graph of j(x) = f (x − 2) will be shifted right two units. j(2) = f (0) We will see that f (x) + c will be a vertical translation. If c is positive, then the graph of f will be shifted up. If c is negative, then the graph of f will be shifted down. Let’s look at an example.

Vertical Translations

We saw that f (x + c) is a horizontal translation. If c is positive, then the graph of f will be shifted up. If c is negative, then the graph of f will be shifted down. Let’s look at an example.

Vertical Translations

We saw that f (x + c) is a horizontal translation. We will see that f (x) + c will be a vertical translation. If c is negative, then the graph of f will be shifted down. Let’s look at an example.

Vertical Translations

We saw that f (x + c) is a horizontal translation. We will see that f (x) + c will be a vertical translation. If c is positive, then the graph of f will be shifted up. Let’s look at an example.

Vertical Translations

We saw that f (x + c) is a horizontal translation. We will see that f (x) + c will be a vertical translation. If c is positive, then the graph of f will be shifted up. If c is negative, then the graph of f will be shifted down. Vertical Translations

We saw that f (x + c) is a horizontal translation. We will see that f (x) + c will be a vertical translation. If c is positive, then the graph of f will be shifted up. If c is negative, then the graph of f will be shifted down. Let’s look at an example. The graph of h(x) = x2 + 3 will be shifted up three units.

Vertical Translations, Example 1 (x 2)

Again we have the graph of f (x) = x2. Vertical Translations, Example 1 (x 2)

Again we have the graph of f (x) = x2.

The graph of h(x) = x2 + 3 will be shifted up three units. Vertical Translations, Example 1 (x 2)

Again we have the graph of f (x) = x2.

The graph of h(x) = x2 + 3 will be shifted up three units. Vertical Translations, Example 1 (x 2)

Again we have the graph of f (x) = x2.

The graph of j(x) = x2 − 3 will be shifted down three units. Vertical Translations, Example 1 (x 2)

Again we have the graph of f (x) = x2.

The graph of j(x) = x2 − 3 will be shifted down three units. The graph of h(x) = f (x) + 2 will be shifted up two units.

Vertical Translations, Example 2

Again we have the graph of some function f . Vertical Translations, Example 2

Again we have the graph of some function f .

The graph of h(x) = f (x) + 2 will be shifted up two units. Vertical Translations, Example 2

Again we have the graph of some function f .

The graph of h(x) = f (x) + 2 will be shifted up two units. Vertical Translations, Example 2

Again we have the graph of some function f .

The graph of j(x) = f (x) − 1 will be shifted down one unit. Vertical Translations, Example 2

Again we have the graph of some function f .

The graph of j(x) = f (x) − 1 will be shifted down one unit. Thus, if we compose f (x) with the function g(x) = −x we will ”flip” or reflect about the x-axis or y-axis. g ◦ f (x) = −f (x) means the points in the range of f are flipped, and we reflect about the x-axis. f ◦ g(x) = f (−x) means the points in the domain of f are flipped. We would reflect about the y-axis. Let’s look at an example.

Reflections

Multiplying by −1 will make negative numbers positive and positive numbers negative. g ◦ f (x) = −f (x) means the points in the range of f are flipped, and we reflect about the x-axis. f ◦ g(x) = f (−x) means the points in the domain of f are flipped. We would reflect about the y-axis. Let’s look at an example.

Reflections

Multiplying by −1 will make negative numbers positive and positive numbers negative. Thus, if we compose f (x) with the function g(x) = −x we will ”flip” or reflect about the x-axis or y-axis. f ◦ g(x) = f (−x) means the points in the domain of f are flipped. We would reflect about the y-axis. Let’s look at an example.

Reflections

Multiplying by −1 will make negative numbers positive and positive numbers negative. Thus, if we compose f (x) with the function g(x) = −x we will ”flip” or reflect about the x-axis or y-axis. g ◦ f (x) = −f (x) means the points in the range of f are flipped, and we reflect about the x-axis. Let’s look at an example.

Reflections

Multiplying by −1 will make negative numbers positive and positive numbers negative. Thus, if we compose f (x) with the function g(x) = −x we will ”flip” or reflect about the x-axis or y-axis. g ◦ f (x) = −f (x) means the points in the range of f are flipped, and we reflect about the x-axis. f ◦ g(x) = f (−x) means the points in the domain of f are flipped. We would reflect about the y-axis. Reflections

Multiplying by −1 will make negative numbers positive and positive numbers negative. Thus, if we compose f (x) with the function g(x) = −x we will ”flip” or reflect about the x-axis or y-axis. g ◦ f (x) = −f (x) means the points in the range of f are flipped, and we reflect about the x-axis. f ◦ g(x) = f (−x) means the points in the domain of f are flipped. We would reflect about the y-axis. Let’s look at an example. h(x) = −(x2) sends the positive y-values to negative y-values and we reflect about the x-axis.

Reflections, Example 1 (x 2)

Let’s look at the graph of f (x) = x2 a third time. Reflections, Example 1 (x 2)

Let’s look at the graph of f (x) = x2 a third time.

h(x) = −(x2) sends the positive y-values to negative y-values and we reflect about the x-axis. Reflections, Example 1 (x 2)

Let’s look at the graph of f (x) = x2 a third time.

h(x) = −(x2) sends the positive y-values to negative y-values and we reflect about the x-axis. Reflections, Example 1 (x 2)

Let’s look at the graph of f (x) = x2 a third time.

j(x) = (−x)2 sends the positive x-values to negative x-values and we reflect about the y-axis. Reflections, Example 1 (x 2)

Let’s look at the graph of f (x) = x2 a third time.

j(x) = (−x)2 sends the positive x-values to negative x-values and we reflect about the y-axis. f (x) = x2 is called an even function.

Reflections, Example 1 (x 2)

Let’s look at the graph of f (x) = x2 a third time.

It shouldn’t be suprising that the graphs of f (x) and f (−x) are the same. Reflections, Example 1 (x 2)

Let’s look at the graph of f (x) = x2 a third time.

It shouldn’t be suprising that the graphs of f (x) and f (−x) are the same. f (x) = x2 is called an even function. h(x) = −f (x) sends the positive y-values to negative y-values and we reflect about the x-axis.

Reflections, Example 2

Let’s look at the graph of f a third time. Reflections, Example 2

Let’s look at the graph of f a third time.

h(x) = −f (x) sends the positive y-values to negative y-values and we reflect about the x-axis. Reflections, Example 2

Let’s look at the graph of f a third time.

h(x) = −f (x) sends the positive y-values to negative y-values and we reflect about the x-axis. Reflections, Example 2

Let’s look at the graph of f a third time.

j(x) = f (−x) sends the positive x-values to negative x-values and we reflect about the y-axis. Reflections, Example 2

Let’s look at the graph of f a third time.

j(x) = f (−x) sends the positive x-values to negative x-values and we reflect about the y-axis. Reflections, Example 2

Let’s look at the graph of f a third time.

−f (x) = f (−x), f (x) is called an odd function. Even and Odd Functions

Definition A function f is even if f (x) = f (−x).

Example x2, cos(x), cos(x) + x2 are even functions.

Definition A function f is odd if −f (x) = f (−x).

Example x3, sin(x), sin(x) + x5 are odd functions. We can immediately see one transformation f (x) = −(x2 + 2x − 8). We can also factor to see that f (x) = −(x − 2)(x + 4), but it is more helpful if we write f (x) = −(x + b)2 + c in order easily apply the transformations. Written as above we can see we will translate horizontally by b, reflect about the x-axis, then translate vertically by c. The process we will use is called completing the square.

Graphing Using Translations

We are capable of graphing new functions if the transformations are given to us, but how can we use the transformations to graph something like f (x) = −x2 − 2x + 8? We can also factor to see that f (x) = −(x − 2)(x + 4), but it is more helpful if we write f (x) = −(x + b)2 + c in order easily apply the transformations. Written as above we can see we will translate horizontally by b, reflect about the x-axis, then translate vertically by c. The process we will use is called completing the square.

Graphing Using Translations

We are capable of graphing new functions if the transformations are given to us, but how can we use the transformations to graph something like f (x) = −x2 − 2x + 8? We can immediately see one transformation f (x) = −(x2 + 2x − 8). Written as above we can see we will translate horizontally by b, reflect about the x-axis, then translate vertically by c. The process we will use is called completing the square.

Graphing Using Translations

We are capable of graphing new functions if the transformations are given to us, but how can we use the transformations to graph something like f (x) = −x2 − 2x + 8? We can immediately see one transformation f (x) = −(x2 + 2x − 8). We can also factor to see that f (x) = −(x − 2)(x + 4), but it is more helpful if we write f (x) = −(x + b)2 + c in order easily apply the transformations. The process we will use is called completing the square.

Graphing Using Translations

We are capable of graphing new functions if the transformations are given to us, but how can we use the transformations to graph something like f (x) = −x2 − 2x + 8? We can immediately see one transformation f (x) = −(x2 + 2x − 8). We can also factor to see that f (x) = −(x − 2)(x + 4), but it is more helpful if we write f (x) = −(x + b)2 + c in order easily apply the transformations. Written as above we can see we will translate horizontally by b, reflect about the x-axis, then translate vertically by c. Graphing Using Translations

We are capable of graphing new functions if the transformations are given to us, but how can we use the transformations to graph something like f (x) = −x2 − 2x + 8? We can immediately see one transformation f (x) = −(x2 + 2x − 8). We can also factor to see that f (x) = −(x − 2)(x + 4), but it is more helpful if we write f (x) = −(x + b)2 + c in order easily apply the transformations. Written as above we can see we will translate horizontally by b, reflect about the x-axis, then translate vertically by c. The process we will use is called completing the square. −x2 − 2x + 8 = −(x2 + 2x − 8) Second, we find b by dividing the coefficient in front of x by 2, so b = 1 and then add 0 by adding and subtract b2. −(x2 + 2x − 8) = −(x2 + 2x + 1 − 1 − 8) Third, factor our perfect square and simplify to find c. −(x2 + 2x − 8) = −(x2 + 2x + 1 − 1 − 8) = −((x + 1)2 − 9) = −(x + 1)2 + 9 To graph we start with x2, shift left 1, reflect down about the x-axis, then shift up 9.

Completing the Square, Example 1 (−x 2 − 2x + 8)

First make the coefficient in front of x2 is 1. Second, we find b by dividing the coefficient in front of x by 2, so b = 1 and then add 0 by adding and subtract b2. −(x2 + 2x − 8) = −(x2 + 2x + 1 − 1 − 8) Third, factor our perfect square and simplify to find c. −(x2 + 2x − 8) = −(x2 + 2x + 1 − 1 − 8) = −((x + 1)2 − 9) = −(x + 1)2 + 9 To graph we start with x2, shift left 1, reflect down about the x-axis, then shift up 9.

Completing the Square, Example 1 (−x 2 − 2x + 8)

First make the coefficient in front of x2 is 1. −x2 − 2x + 8 = −(x2 + 2x − 8) −(x2 + 2x − 8) = −(x2 + 2x + 1 − 1 − 8) Third, factor our perfect square and simplify to find c. −(x2 + 2x − 8) = −(x2 + 2x + 1 − 1 − 8) = −((x + 1)2 − 9) = −(x + 1)2 + 9 To graph we start with x2, shift left 1, reflect down about the x-axis, then shift up 9.

Completing the Square, Example 1 (−x 2 − 2x + 8)

First make the coefficient in front of x2 is 1. −x2 − 2x + 8 = −(x2 + 2x − 8) Second, we find b by dividing the coefficient in front of x by 2, so b = 1 and then add 0 by adding and subtract b2. Third, factor our perfect square and simplify to find c. −(x2 + 2x − 8) = −(x2 + 2x + 1 − 1 − 8) = −((x + 1)2 − 9) = −(x + 1)2 + 9 To graph we start with x2, shift left 1, reflect down about the x-axis, then shift up 9.

Completing the Square, Example 1 (−x 2 − 2x + 8)

First make the coefficient in front of x2 is 1. −x2 − 2x + 8 = −(x2 + 2x − 8) Second, we find b by dividing the coefficient in front of x by 2, so b = 1 and then add 0 by adding and subtract b2. −(x2 + 2x − 8) = −(x2 + 2x + 1 − 1 − 8) −(x2 + 2x − 8) = −(x2 + 2x + 1 − 1 − 8) = −((x + 1)2 − 9) = −(x + 1)2 + 9 To graph we start with x2, shift left 1, reflect down about the x-axis, then shift up 9.

Completing the Square, Example 1 (−x 2 − 2x + 8)

First make the coefficient in front of x2 is 1. −x2 − 2x + 8 = −(x2 + 2x − 8) Second, we find b by dividing the coefficient in front of x by 2, so b = 1 and then add 0 by adding and subtract b2. −(x2 + 2x − 8) = −(x2 + 2x + 1 − 1 − 8) Third, factor our perfect square and simplify to find c. To graph we start with x2, shift left 1, reflect down about the x-axis, then shift up 9.

Completing the Square, Example 1 (−x 2 − 2x + 8)

First make the coefficient in front of x2 is 1. −x2 − 2x + 8 = −(x2 + 2x − 8) Second, we find b by dividing the coefficient in front of x by 2, so b = 1 and then add 0 by adding and subtract b2. −(x2 + 2x − 8) = −(x2 + 2x + 1 − 1 − 8) Third, factor our perfect square and simplify to find c. −(x2 + 2x − 8) = −(x2 + 2x + 1 − 1 − 8) = −((x + 1)2 − 9) = −(x + 1)2 + 9 Completing the Square, Example 1 (−x 2 − 2x + 8)

First make the coefficient in front of x2 is 1. −x2 − 2x + 8 = −(x2 + 2x − 8) Second, we find b by dividing the coefficient in front of x by 2, so b = 1 and then add 0 by adding and subtract b2. −(x2 + 2x − 8) = −(x2 + 2x + 1 − 1 − 8) Third, factor our perfect square and simplify to find c. −(x2 + 2x − 8) = −(x2 + 2x + 1 − 1 − 8) = −((x + 1)2 − 9) = −(x + 1)2 + 9 To graph we start with x2, shift left 1, reflect down about the x-axis, then shift up 9. 2 2 b c ax + bx + c = a(x + a + a ) b Find the horizontal shift by dividing a by 2 and keep equality by adding 0 cleverly. 2 b c b b 2 b 2 c a(x + a + a ) = a(x + a + 2a − 2a + a ). Factor and simplify b b 2 b 2 c b 2 c b 2 b 2 b2 a(x+ a + 2a − 2a + a ) = a((x+ 2a ) + a − 2a ) = a(x+ 2a ) +c− 4a .

Completing the Square, Example 2 (ax 2 + bx + c)

Again, step 1 is factor out the coefficient a b Find the horizontal shift by dividing a by 2 and keep equality by adding 0 cleverly. 2 b c b b 2 b 2 c a(x + a + a ) = a(x + a + 2a − 2a + a ). Factor and simplify b b 2 b 2 c b 2 c b 2 b 2 b2 a(x+ a + 2a − 2a + a ) = a((x+ 2a ) + a − 2a ) = a(x+ 2a ) +c− 4a .

Completing the Square, Example 2 (ax 2 + bx + c)

Again, step 1 is factor out the coefficient a 2 2 b c ax + bx + c = a(x + a + a ) 2 b c b b 2 b 2 c a(x + a + a ) = a(x + a + 2a − 2a + a ). Factor and simplify b b 2 b 2 c b 2 c b 2 b 2 b2 a(x+ a + 2a − 2a + a ) = a((x+ 2a ) + a − 2a ) = a(x+ 2a ) +c− 4a .

Completing the Square, Example 2 (ax 2 + bx + c)

Again, step 1 is factor out the coefficient a 2 2 b c ax + bx + c = a(x + a + a ) b Find the horizontal shift by dividing a by 2 and keep equality by adding 0 cleverly. Factor and simplify b b 2 b 2 c b 2 c b 2 b 2 b2 a(x+ a + 2a − 2a + a ) = a((x+ 2a ) + a − 2a ) = a(x+ 2a ) +c− 4a .

Completing the Square, Example 2 (ax 2 + bx + c)

Again, step 1 is factor out the coefficient a 2 2 b c ax + bx + c = a(x + a + a ) b Find the horizontal shift by dividing a by 2 and keep equality by adding 0 cleverly. 2 b c b b 2 b 2 c a(x + a + a ) = a(x + a + 2a − 2a + a ). Completing the Square, Example 2 (ax 2 + bx + c)

Again, step 1 is factor out the coefficient a 2 2 b c ax + bx + c = a(x + a + a ) b Find the horizontal shift by dividing a by 2 and keep equality by adding 0 cleverly. 2 b c b b 2 b 2 c a(x + a + a ) = a(x + a + 2a − 2a + a ). Factor and simplify b b 2 b 2 c b 2 c b 2 b 2 b2 a(x+ a + 2a − 2a + a ) = a((x+ 2a ) + a − 2a ) = a(x+ 2a ) +c− 4a . There is a nice legend about the payment of wheat and a chessboard that illustrates this quite nicely. Question What is the one point that will agree on all graphs of exponential functions bx ?

Question What can you say about exponentional functions with b < 1?

Exponential Functions

Definition Let b > 0, an exponential function is a function of the form

f (x) = bx

where b is the base, and x is the exponent.

Exponential functions with b > 1 grow incredibly fast, faster than any polynomial. Question What is the one point that will agree on all graphs of exponential functions bx ?

Question What can you say about exponentional functions with b < 1?

Exponential Functions

Definition Let b > 0, an exponential function is a function of the form

f (x) = bx

where b is the base, and x is the exponent.

Exponential functions with b > 1 grow incredibly fast, faster than any polynomial. There is a nice legend about the payment of wheat and a chessboard that illustrates this quite nicely. Question What can you say about exponentional functions with b < 1?

Exponential Functions

Definition Let b > 0, an exponential function is a function of the form

f (x) = bx

where b is the base, and x is the exponent.

Exponential functions with b > 1 grow incredibly fast, faster than any polynomial. There is a nice legend about the payment of wheat and a chessboard that illustrates this quite nicely. Question What is the one point that will agree on all graphs of exponential functions bx ? Exponential Functions

Definition Let b > 0, an exponential function is a function of the form

f (x) = bx

where b is the base, and x is the exponent.

Exponential functions with b > 1 grow incredibly fast, faster than any polynomial. There is a nice legend about the payment of wheat and a chessboard that illustrates this quite nicely. Question What is the one point that will agree on all graphs of exponential functions bx ?

Question What can you say about exponentional functions with b < 1? Exponential Functions, a Question

Question Let f be a continuous function such that f (x + y) = f (x)f (y), and f (1) = 2. (a) What are f (2) and f (3)? If n is an integer, what is f (n)? m m (b) For a rational number n , what is f ( n )? We will see in the section on continuity that it is ”enough” to uniquely define a continuous function on the rationals. Laws of Exponents

Theorem For a, b, x, y ∈ R, a, b > 0 ax+y = ax ay x−y ax a = ay (ax )y = axy (ab)x = ax bx

Question Show that the second and thirds laws follow from the first. Graphing Exponentials

Question

Which graph is the graph of the exponential with the largest base? (for ax , a is called the base) Question How can we write an exponential with base a < 1 as a composition of an exponential with base b > 1 and a transformation?

Let’s look at an example

Graphing Exponentials

Exponentials functions are affected by transformations just like any other function. Let’s look at an example

Graphing Exponentials

Exponentials functions are affected by transformations just like any other function. Question How can we write an exponential with base a < 1 as a composition of an exponential with base b > 1 and a transformation? Graphing Exponentials

Exponentials functions are affected by transformations just like any other function. Question How can we write an exponential with base a < 1 as a composition of an exponential with base b > 1 and a transformation?

Let’s look at an example 3
x Graphing Exponentials, Example 1 f (x) = 2

Let’s look at the base graph 3
x Graphing Exponentials, Example 1 f (x) = 2

Let’s look at the base graph 3
x Graphing Exponentials, Example 1 f (x) = 2

Let’s look at the base graph

The two most important points are (0, f (0)) and (1, f (1)), those seems right for the sketch. 3
x Graphing Exponentials, Example 1 f (x) = 2

Let’s look at the base graph

We can shift it vertically by 1. Giving us 3
x g(x) = f (x) + 1 = 2 + 1.

3
x Graphing Exponentials, Example 1 f (x) = 2

Let’s look at the base graph

We can shift it vertically by 1. 3
x Graphing Exponentials, Example 1 f (x) = 2

Let’s look at the base graph

We can shift it vertically by 1. Giving us 3
x g(x) = f (x) + 1 = 2 + 1. 3
x Graphing Exponentials, Example 1 f (x) = 2

Let’s look at the base graph

We can shift it by 1 and horizontally by 1. Giving us 3
(x−1) h(x) = f (x − 1) + 1 = 2 + 1.

3
x Graphing Exponentials, Example 1 f (x) = 2

Let’s look at the base graph

We can shift it by 1 and horizontally by 1. 3
x Graphing Exponentials, Example 1 f (x) = 2

Let’s look at the base graph

We can shift it by 1 and horizontally by 1. Giving us 3
(x−1) h(x) = f (x − 1) + 1 = 2 + 1. 3
x Graphing Exponentials, Example 1 f (x) = 2

Let’s look at the base graph

We can also reflect it about the y-axis. 3
−x 2
x Giving us g(x) = f (−x) = 2 = 3 , which should answer the preceding question.

3
x Graphing Exponentials, Example 1 f (x) = 2

Let’s look at the base graph

We can also reflect it about the y-axis. 3
x Graphing Exponentials, Example 1 f (x) = 2

Let’s look at the base graph

We can also reflect it about the y-axis. 3
−x 2
x Giving us g(x) = f (−x) = 2 = 3 , which should answer the preceding question. √ 3g(1) = g(2) √ 3Ca = Ca2 √ 3 = a √ C( 3)2 = 9 C = 3

Identifying Exponentials Algebraically

Example x Find constants√ C, a > 0 such that g(x) = Ca such that g(1) = 3 3, g(2) = 9, g(4) = 27. √ 3Ca = Ca2 √ 3 = a √ C( 3)2 = 9 C = 3

Identifying Exponentials Algebraically

Example x Find constants√ C, a > 0 such that g(x) = Ca such that g(1) = 3 3, g(2) = 9, g(4) = 27.

√ 3g(1) = g(2) √ 3 = a √ C( 3)2 = 9 C = 3

Identifying Exponentials Algebraically

Example x Find constants√ C, a > 0 such that g(x) = Ca such that g(1) = 3 3, g(2) = 9, g(4) = 27.

√ 3g(1) = g(2) √ 3Ca = Ca2 √ C( 3)2 = 9 C = 3

Identifying Exponentials Algebraically

Example x Find constants√ C, a > 0 such that g(x) = Ca such that g(1) = 3 3, g(2) = 9, g(4) = 27.

√ 3g(1) = g(2) √ 3Ca = Ca2 √ 3 = a C = 3

Identifying Exponentials Algebraically

Example x Find constants√ C, a > 0 such that g(x) = Ca such that g(1) = 3 3, g(2) = 9, g(4) = 27.

√ 3g(1) = g(2) √ 3Ca = Ca2 √ 3 = a √ C( 3)2 = 9 Identifying Exponentials Algebraically

Example x Find constants√ C, a > 0 such that g(x) = Ca such that g(1) = 3 3, g(2) = 9, g(4) = 27.

√ 3g(1) = g(2) √ 3Ca = Ca2 √ 3 = a √ C( 3)2 = 9 C = 3 Definition

We will define the logarithm with base a as the function loga(x) that satisfies

x loga (a ) = x aloga x = x

Question x log x What is the domain for loga (a )? a a ? (Where are these identities valid?)

Logarithms, the Inverse of Exponentials

The title says that logarithms are inverses of exponential functions, but what does that mean? Question x log x What is the domain for loga (a )? a a ? (Where are these identities valid?)

Logarithms, the Inverse of Exponentials

The title says that logarithms are inverses of exponential functions, but what does that mean? Definition

We will define the logarithm with base a as the function loga(x) that satisfies

x loga (a ) = x aloga x = x Logarithms, the Inverse of Exponentials

The title says that logarithms are inverses of exponential functions, but what does that mean? Definition

We will define the logarithm with base a as the function loga(x) that satisfies

x loga (a ) = x aloga x = x

Question x log x What is the domain for loga (a )? a a ? (Where are these identities valid?) Logarithms, the Inverse of Exponentials

The definition gives us some very important properties of logarithms and their relations to exponentials. Theorem y loga x = y ⇔ a = x

loga(xy) = loga(x) + loga(y)  x  loga y = loga(x) − loga(y) r loga (x ) = r loga(x) log (x) = logb(x) a logb(a)

Question Use the definition and properties of exponents to prove the Log Laws. 81 1 = log 81 − log 27 = log x x x 27 1 = logx 3 ⇒x1 = 3 x = 3

Using Logarithms to Evaluate

Example

Solve logx 81 − logx 27 = 1 for x. 1 = logx 3 ⇒x1 = 3 x = 3

Using Logarithms to Evaluate

Example

Solve logx 81 − logx 27 = 1 for x.

81 1 = log 81 − log 27 = log x x x 27 ⇒x1 = 3 x = 3

Using Logarithms to Evaluate

Example

Solve logx 81 − logx 27 = 1 for x.

81 1 = log 81 − log 27 = log x x x 27 1 = logx 3 x = 3

Using Logarithms to Evaluate

Example

Solve logx 81 − logx 27 = 1 for x.

81 1 = log 81 − log 27 = log x x x 27 1 = logx 3 ⇒x1 = 3 Using Logarithms to Evaluate

Example

Solve logx 81 − logx 27 = 1 for x.

81 1 = log 81 − log 27 = log x x x 27 1 = logx 3 ⇒x1 = 3 x = 3 Example Rewrite 2x as ekx Recall that eln x = x, so we can rewrite 2 as eln 2. Now substitute, 2x = (eln 2)x = ex ln 2

Using Logarithms to Evaluate

We will see during the section on derivatives that rewriting exponentials as powers of e is very useful. The key to rewriting exponentials as powers of e is in the natural logarithm ln = loge Recall that eln x = x, so we can rewrite 2 as eln 2. Now substitute, 2x = (eln 2)x = ex ln 2

Using Logarithms to Evaluate

We will see during the section on derivatives that rewriting exponentials as powers of e is very useful. The key to rewriting exponentials as powers of e is in the natural logarithm ln = loge Example Rewrite 2x as ekx Now substitute, 2x = (eln 2)x = ex ln 2

Using Logarithms to Evaluate

We will see during the section on derivatives that rewriting exponentials as powers of e is very useful. The key to rewriting exponentials as powers of e is in the natural logarithm ln = loge Example Rewrite 2x as ekx Recall that eln x = x, so we can rewrite 2 as eln 2. Using Logarithms to Evaluate

We will see during the section on derivatives that rewriting exponentials as powers of e is very useful. The key to rewriting exponentials as powers of e is in the natural logarithm ln = loge Example Rewrite 2x as ekx Recall that eln x = x, so we can rewrite 2 as eln 2. Now substitute, 2x = (eln 2)x = ex ln 2 x −1 2 −1 √ f (x) = b , f (y) = logb y and g(x) = x , g (y) = y We have also seen that f (f −1(x)) = x = f −1(f (x)) and g(g −1(x)) = x = g −1(g(x)) where the compositions are defined y We can generalize the Log Law logb x = y ⇔ b = x as Theorem f (x) = y ⇔ f −1(y) = x.

Inverses

We have seen two examples of inverses already. We have also seen that f (f −1(x)) = x = f −1(f (x)) and g(g −1(x)) = x = g −1(g(x)) where the compositions are defined y We can generalize the Log Law logb x = y ⇔ b = x as Theorem f (x) = y ⇔ f −1(y) = x.

Inverses

We have seen two examples of inverses already. x −1 2 −1 √ f (x) = b , f (y) = logb y and g(x) = x , g (y) = y y We can generalize the Log Law logb x = y ⇔ b = x as Theorem f (x) = y ⇔ f −1(y) = x.

Inverses

We have seen two examples of inverses already. x −1 2 −1 √ f (x) = b , f (y) = logb y and g(x) = x , g (y) = y We have also seen that f (f −1(x)) = x = f −1(f (x)) and g(g −1(x)) = x = g −1(g(x)) where the compositions are defined Inverses

We have seen two examples of inverses already. x −1 2 −1 √ f (x) = b , f (y) = logb y and g(x) = x , g (y) = y We have also seen that f (f −1(x)) = x = f −1(f (x)) and g(g −1(x)) = x = g −1(g(x)) where the compositions are defined y We can generalize the Log Law logb x = y ⇔ b = x as Theorem f (x) = y ⇔ f −1(y) = x. Questions Basic Skills

Question Find the domain of √ x x+1

Question Let f (x) = 3x + 2, find d so that f (x + 4) = f (x) + d. What does this tell you about translations of linear functions. Questions Understanding

Question Find a function whose domain is bounded, contained in an interval (a, b), but whose range is unbounded.

Question Find a function whose domain is bounded, but whose range is all of R. Question We have seen that we can translate and reflect graphs. Can we rotate them? Why or Why not?

Question If the domain of f (x) is (−4, 2), then what is the domain of f (x + c)? Questions Understanding

Question If f (x) is an odd function, then what must f (0) equal?

Question Why do we write a general exponentional f (x) = Cax instead of f (x) = Capx ?