Elementary Functions Part 1, Functions Lecture 1.6d, Function Inverses: One-to-one and onto functions
Dr. Ken W. Smith
Sam Houston State University
2013
Smith (SHSU) Elementary Functions 2013 26 / 33 Function Inverses
When does a function f have an inverse? It turns out that there are two critical properties necessary for a function f to be invertible. The function needs to be “one-to-one” and “onto”.
Smith (SHSU) Elementary Functions 2013 27 / 33 Function Inverses
When does a function f have an inverse? It turns out that there are two critical properties necessary for a function f to be invertible. The function needs to be “one-to-one” and “onto”.
Smith (SHSU) Elementary Functions 2013 27 / 33 Function Inverses
When does a function f have an inverse? It turns out that there are two critical properties necessary for a function f to be invertible. The function needs to be “one-to-one” and “onto”.
Smith (SHSU) Elementary Functions 2013 27 / 33 Function Inverses
When does a function f have an inverse? It turns out that there are two critical properties necessary for a function f to be invertible. The function needs to be “one-to-one” and “onto”.
Smith (SHSU) Elementary Functions 2013 27 / 33 One-to-one functions*
A function like f(x) = x2 maps two different inputs, x = −5 and x = 5, to the same output, y = 25. But with a one-to-one function no pair of inputs give the same output.
Here is a function we saw earlier. This function is not one-to-one since both the inputs x = 2 and x = 3 give the output y = C.
Smith (SHSU) Elementary Functions 2013 28 / 33 One-to-one functions*
A function like f(x) = x2 maps two different inputs, x = −5 and x = 5, to the same output, y = 25. But with a one-to-one function no pair of inputs give the same output.
Here is a function we saw earlier. This function is not one-to-one since both the inputs x = 2 and x = 3 give the output y = C.
Smith (SHSU) Elementary Functions 2013 28 / 33 One-to-one functions*
A function like f(x) = x2 maps two different inputs, x = −5 and x = 5, to the same output, y = 25. But with a one-to-one function no pair of inputs give the same output.
Here is a function we saw earlier. This function is not one-to-one since both the inputs x = 2 and x = 3 give the output y = C.
Smith (SHSU) Elementary Functions 2013 28 / 33 One-to-one functions*
A function like f(x) = x2 maps two different inputs, x = −5 and x = 5, to the same output, y = 25. But with a one-to-one function no pair of inputs give the same output.
Here is a function we saw earlier. This function is not one-to-one since both the inputs x = 2 and x = 3 give the output y = C.
Smith (SHSU) Elementary Functions 2013 28 / 33 Function Inverses
We can define one-to-one function more formally, as a mathematical implication:
A function f is one-to-one if f(a) = f(b) =⇒ a = b.
One can think of this implication this way: “A function is one-to-one if whenever it appears that two inputs a and b give the same output, in fact a and b were the same.” Let’s show that the function f(x) = 3x + 5 defined earlier is one-to-one. We do this is a series of steps:
1 Suppose that f(a) = f(b). 2 Then 3a + 5 = 3b + 5. 3 Subtract 5 from both sides to get 3a = 3b. 4 Divide both sides by 3 to get a = b. Since we have shown that f(a) = f(b) =⇒ a = b, the function f(x) is one-to-one.
Smith (SHSU) Elementary Functions 2013 29 / 33 Function Inverses
We can define one-to-one function more formally, as a mathematical implication:
A function f is one-to-one if f(a) = f(b) =⇒ a = b.
One can think of this implication this way: “A function is one-to-one if whenever it appears that two inputs a and b give the same output, in fact a and b were the same.” Let’s show that the function f(x) = 3x + 5 defined earlier is one-to-one. We do this is a series of steps:
1 Suppose that f(a) = f(b). 2 Then 3a + 5 = 3b + 5. 3 Subtract 5 from both sides to get 3a = 3b. 4 Divide both sides by 3 to get a = b. Since we have shown that f(a) = f(b) =⇒ a = b, the function f(x) is one-to-one.
Smith (SHSU) Elementary Functions 2013 29 / 33 Function Inverses
We can define one-to-one function more formally, as a mathematical implication:
A function f is one-to-one if f(a) = f(b) =⇒ a = b.
One can think of this implication this way: “A function is one-to-one if whenever it appears that two inputs a and b give the same output, in fact a and b were the same.” Let’s show that the function f(x) = 3x + 5 defined earlier is one-to-one. We do this is a series of steps:
1 Suppose that f(a) = f(b). 2 Then 3a + 5 = 3b + 5. 3 Subtract 5 from both sides to get 3a = 3b. 4 Divide both sides by 3 to get a = b. Since we have shown that f(a) = f(b) =⇒ a = b, the function f(x) is one-to-one.
Smith (SHSU) Elementary Functions 2013 29 / 33 Function Inverses
We can define one-to-one function more formally, as a mathematical implication:
A function f is one-to-one if f(a) = f(b) =⇒ a = b.
One can think of this implication this way: “A function is one-to-one if whenever it appears that two inputs a and b give the same output, in fact a and b were the same.” Let’s show that the function f(x) = 3x + 5 defined earlier is one-to-one. We do this is a series of steps:
1 Suppose that f(a) = f(b). 2 Then 3a + 5 = 3b + 5. 3 Subtract 5 from both sides to get 3a = 3b. 4 Divide both sides by 3 to get a = b. Since we have shown that f(a) = f(b) =⇒ a = b, the function f(x) is one-to-one.
Smith (SHSU) Elementary Functions 2013 29 / 33 Function Inverses
We can define one-to-one function more formally, as a mathematical implication:
A function f is one-to-one if f(a) = f(b) =⇒ a = b.
One can think of this implication this way: “A function is one-to-one if whenever it appears that two inputs a and b give the same output, in fact a and b were the same.” Let’s show that the function f(x) = 3x + 5 defined earlier is one-to-one. We do this is a series of steps:
1 Suppose that f(a) = f(b). 2 Then 3a + 5 = 3b + 5. 3 Subtract 5 from both sides to get 3a = 3b. 4 Divide both sides by 3 to get a = b. Since we have shown that f(a) = f(b) =⇒ a = b, the function f(x) is one-to-one.
Smith (SHSU) Elementary Functions 2013 29 / 33 Function Inverses
We can define one-to-one function more formally, as a mathematical implication:
A function f is one-to-one if f(a) = f(b) =⇒ a = b.
One can think of this implication this way: “A function is one-to-one if whenever it appears that two inputs a and b give the same output, in fact a and b were the same.” Let’s show that the function f(x) = 3x + 5 defined earlier is one-to-one. We do this is a series of steps:
1 Suppose that f(a) = f(b). 2 Then 3a + 5 = 3b + 5. 3 Subtract 5 from both sides to get 3a = 3b. 4 Divide both sides by 3 to get a = b. Since we have shown that f(a) = f(b) =⇒ a = b, the function f(x) is one-to-one.
Smith (SHSU) Elementary Functions 2013 29 / 33 Function Inverses
We can define one-to-one function more formally, as a mathematical implication:
A function f is one-to-one if f(a) = f(b) =⇒ a = b.
One can think of this implication this way: “A function is one-to-one if whenever it appears that two inputs a and b give the same output, in fact a and b were the same.” Let’s show that the function f(x) = 3x + 5 defined earlier is one-to-one. We do this is a series of steps:
1 Suppose that f(a) = f(b). 2 Then 3a + 5 = 3b + 5. 3 Subtract 5 from both sides to get 3a = 3b. 4 Divide both sides by 3 to get a = b. Since we have shown that f(a) = f(b) =⇒ a = b, the function f(x) is one-to-one.
Smith (SHSU) Elementary Functions 2013 29 / 33 Function Inverses
We can define one-to-one function more formally, as a mathematical implication:
A function f is one-to-one if f(a) = f(b) =⇒ a = b.
One can think of this implication this way: “A function is one-to-one if whenever it appears that two inputs a and b give the same output, in fact a and b were the same.” Let’s show that the function f(x) = 3x + 5 defined earlier is one-to-one. We do this is a series of steps:
1 Suppose that f(a) = f(b). 2 Then 3a + 5 = 3b + 5. 3 Subtract 5 from both sides to get 3a = 3b. 4 Divide both sides by 3 to get a = b. Since we have shown that f(a) = f(b) =⇒ a = b, the function f(x) is one-to-one.
Smith (SHSU) Elementary Functions 2013 29 / 33 Function Inverses
We can define one-to-one function more formally, as a mathematical implication:
A function f is one-to-one if f(a) = f(b) =⇒ a = b.
One can think of this implication this way: “A function is one-to-one if whenever it appears that two inputs a and b give the same output, in fact a and b were the same.” Let’s show that the function f(x) = 3x + 5 defined earlier is one-to-one. We do this is a series of steps:
1 Suppose that f(a) = f(b). 2 Then 3a + 5 = 3b + 5. 3 Subtract 5 from both sides to get 3a = 3b. 4 Divide both sides by 3 to get a = b. Since we have shown that f(a) = f(b) =⇒ a = b, the function f(x) is one-to-one.
Smith (SHSU) Elementary Functions 2013 29 / 33 Function Inverses
We can define one-to-one function more formally, as a mathematical implication:
A function f is one-to-one if f(a) = f(b) =⇒ a = b.
One can think of this implication this way: “A function is one-to-one if whenever it appears that two inputs a and b give the same output, in fact a and b were the same.” Let’s show that the function f(x) = 3x + 5 defined earlier is one-to-one. We do this is a series of steps:
1 Suppose that f(a) = f(b). 2 Then 3a + 5 = 3b + 5. 3 Subtract 5 from both sides to get 3a = 3b. 4 Divide both sides by 3 to get a = b. Since we have shown that f(a) = f(b) =⇒ a = b, the function f(x) is one-to-one.
Smith (SHSU) Elementary Functions 2013 29 / 33 Function Inverses
We can define one-to-one function more formally, as a mathematical implication:
A function f is one-to-one if f(a) = f(b) =⇒ a = b.
One can think of this implication this way: “A function is one-to-one if whenever it appears that two inputs a and b give the same output, in fact a and b were the same.” Let’s show that the function f(x) = 3x + 5 defined earlier is one-to-one. We do this is a series of steps:
1 Suppose that f(a) = f(b). 2 Then 3a + 5 = 3b + 5. 3 Subtract 5 from both sides to get 3a = 3b. 4 Divide both sides by 3 to get a = b. Since we have shown that f(a) = f(b) =⇒ a = b, the function f(x) is one-to-one.
Smith (SHSU) Elementary Functions 2013 29 / 33 Other common terms for one-to-one functions.
A one-to-one function is sometimes called an injection or an injective function. Wikipedia uses the term “injective function” in the article here.
Smith (SHSU) Elementary Functions 2013 30 / 33 Other common terms for one-to-one functions.
A one-to-one function is sometimes called an injection or an injective function. Wikipedia uses the term “injective function” in the article here.
Smith (SHSU) Elementary Functions 2013 30 / 33 Onto functions*
A function f : D → C is onto if every possible output in C is truly an output, that is, if the range of f is equal to the entire codomain C.
In the figure above, the function is not onto because the element B is the codomain is not an output. (The element A is also not an output of the function.)
The codomain is Y = {A, B, C, D} but the range is just {C,D}. If f(x) = y then we say that y is the image of x and that x is the preimage of y. If a function f : D → C is onto then every element of the codomain has a preimage.
Smith (SHSU) Elementary Functions 2013 31 / 33 Onto functions*
A function f : D → C is onto if every possible output in C is truly an output, that is, if the range of f is equal to the entire codomain C.
In the figure above, the function is not onto because the element B is the codomain is not an output. (The element A is also not an output of the function.)
The codomain is Y = {A, B, C, D} but the range is just {C,D}. If f(x) = y then we say that y is the image of x and that x is the preimage of y. If a function f : D → C is onto then every element of the codomain has a preimage.
Smith (SHSU) Elementary Functions 2013 31 / 33 Onto functions*
A function f : D → C is onto if every possible output in C is truly an output, that is, if the range of f is equal to the entire codomain C.
In the figure above, the function is not onto because the element B is the codomain is not an output. (The element A is also not an output of the function.)
The codomain is Y = {A, B, C, D} but the range is just {C,D}. If f(x) = y then we say that y is the image of x and that x is the preimage of y. If a function f : D → C is onto then every element of the codomain has a preimage.
Smith (SHSU) Elementary Functions 2013 31 / 33 Onto functions*
A function f : D → C is onto if every possible output in C is truly an output, that is, if the range of f is equal to the entire codomain C.
In the figure above, the function is not onto because the element B is the codomain is not an output. (The element A is also not an output of the function.)
The codomain is Y = {A, B, C, D} but the range is just {C,D}. If f(x) = y then we say that y is the image of x and that x is the preimage of y. If a function f : D → C is onto then every element of the codomain has a preimage.
Smith (SHSU) Elementary Functions 2013 31 / 33 Onto functions*
A function f : D → C is onto if every possible output in C is truly an output, that is, if the range of f is equal to the entire codomain C.
In the figure above, the function is not onto because the element B is the codomain is not an output. (The element A is also not an output of the function.)
The codomain is Y = {A, B, C, D} but the range is just {C,D}. If f(x) = y then we say that y is the image of x and that x is the preimage of y. If a function f : D → C is onto then every element of the codomain has a preimage.
Smith (SHSU) Elementary Functions 2013 31 / 33 Onto functions*
A function f : D → C is onto if every possible output in C is truly an output, that is, if the range of f is equal to the entire codomain C.
In the figure above, the function is not onto because the element B is the codomain is not an output. (The element A is also not an output of the function.)
The codomain is Y = {A, B, C, D} but the range is just {C,D}. If f(x) = y then we say that y is the image of x and that x is the preimage of y. If a function f : D → C is onto then every element of the codomain has a preimage.
Smith (SHSU) Elementary Functions 2013 31 / 33 Onto functions*
A function f : D → C is onto if every possible output in C is truly an output, that is, if the range of f is equal to the entire codomain C.
In the figure above, the function is not onto because the element B is the codomain is not an output. (The element A is also not an output of the function.)
The codomain is Y = {A, B, C, D} but the range is just {C,D}. If f(x) = y then we say that y is the image of x and that x is the preimage of y. If a function f : D → C is onto then every element of the codomain has a preimage.
Smith (SHSU) Elementary Functions 2013 31 / 33 Other common terms for onto functions.
An onto function is sometimes called a surjection or a surjective function. Wikipedia uses the term “surjective function” in the article here.
Smith (SHSU) Elementary Functions 2013 32 / 33 Other common terms for onto functions.
An onto function is sometimes called a surjection or a surjective function. Wikipedia uses the term “surjective function” in the article here.
Smith (SHSU) Elementary Functions 2013 32 / 33 Inverse functions
A function f : D → C has an inverse f −1 : D → C if and only if f is both a one-to-one function and an onto function.
If the function f is onto then every element of C has at least one preimage back in D. If the function f is one-to-one then every element of C which has a preimage has a unique preimage. (Recall that this uniqueness is the critical part of the definition of a function!)
(END)
Smith (SHSU) Elementary Functions 2013 33 / 33 Inverse functions
A function f : D → C has an inverse f −1 : D → C if and only if f is both a one-to-one function and an onto function.
If the function f is onto then every element of C has at least one preimage back in D. If the function f is one-to-one then every element of C which has a preimage has a unique preimage. (Recall that this uniqueness is the critical part of the definition of a function!)
(END)
Smith (SHSU) Elementary Functions 2013 33 / 33 Inverse functions
A function f : D → C has an inverse f −1 : D → C if and only if f is both a one-to-one function and an onto function.
If the function f is onto then every element of C has at least one preimage back in D. If the function f is one-to-one then every element of C which has a preimage has a unique preimage. (Recall that this uniqueness is the critical part of the definition of a function!)
(END)
Smith (SHSU) Elementary Functions 2013 33 / 33 Inverse functions
A function f : D → C has an inverse f −1 : D → C if and only if f is both a one-to-one function and an onto function.
If the function f is onto then every element of C has at least one preimage back in D. If the function f is one-to-one then every element of C which has a preimage has a unique preimage. (Recall that this uniqueness is the critical part of the definition of a function!)
(END)
Smith (SHSU) Elementary Functions 2013 33 / 33 Inverse functions
A function f : D → C has an inverse f −1 : D → C if and only if f is both a one-to-one function and an onto function.
If the function f is onto then every element of C has at least one preimage back in D. If the function f is one-to-one then every element of C which has a preimage has a unique preimage. (Recall that this uniqueness is the critical part of the definition of a function!)
(END)
Smith (SHSU) Elementary Functions 2013 33 / 33