Chapter 3 Functions and Their Graphs
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Chapter 3 Functions and Their Graphs 3.1 Functions 1. Determine Whether a Relation Represents a Function Definition A relation is a correspondence between two sets. If a relation exists between x and y, then we say that x corresponds to y or that y depends on x, and we write x→y, or (x, y).
Definition Let X and Y be two nonempty sets. A function from X into Y is a relation that associates with each element of X exactly one element of Y.
The set X is called the domain of the function. For each element x in X, the corresponding element y in Y is called the value of the function at x, or the image of x. The set of all images of the elements of the domain is called the range of the function. Note that not all relations between two sets are functions.
Example 1 Determine whether each relation represents a function. If it is a function, state the domain and range.
1 Example 2 Determine whether each relation represents a function. If it is a function, state the domain and range (a) (1,4),(2,5),(3,6),(4,7)
(b) (1,4),(2,4),(3,5),(6,10)
(c) (3,9),(2,4),(0,0),(1,1),(3,8)
Example 3 Determine whether the equation y 2x 5 defines y as a function of x.
Example 4 Determine whether the equation x 2 y 2 1 defines y as a function of x.
2. Find the Value of a Function If f is a function, then for each number x in its domain the corresponding image in the range is denoted by f (x) , read as “f of x” or “f at x”. f (x) is the value of f at the number x.
2 Example 5 For the function f (x) 2x 2 3x , evaluate (a) f (3) , (b) f (x) f (3) , (c) f (x) , (d) f (x) , (e) f (x 3) , f (x h) f (x) (f) , h 0 h
3. Find the Domain of a Function The domain of a function is the largest set of real numbers for which the value f (x) is a real number.
Example 6 Find the domain of each of the following functions: (a) f (x) x 2 5x
3x (b) g(x) x 2 4
3 (c) h(t) 4 3t
4. Form the Sum, Difference, Product, and Quotient of Two Functions ( f g)(x) f (x) g(x)
( f g)(x) f (x) g(x)
( f g)(x) f (x) g(x)
f f (x) (x) g g(x)
Example 7 Let f and g be two functions defined as f (x) 2x 2 3 and g(x) 4x 3 1. Find the following, and determine the domain in each case. f (a) ( f g)(x) , (b) ( f g)(x) , (c) ( f g)(x) , (d) (x) g
4 3.2 The Graph of a Function 1. Identify the Graph of a Function Vertical-line Test: A set of points in the xy-plane is the graph of a function if and only if every vertical line intersects the graph in at most one point.
Example1 Which of the graphs are graphs of functions?
2. Obtain Information from or about the Graph of a Function If (x, y) is a point on the graph of a function f, the y is the value of f at x; that is y = f (x). In other words, point (x, f (x)) is always on the graph of function f for ach x in its domain.
Example2 Let f be the function whose graph is given. 3 (a) What are f (0), f and f (3 ) ? 2 (b) What is the domain of f ? (c) What is the range of f ? (d) List the intercepts. (e) How often does the line y 2 intersect the graph? (f) For what values of x does f (x) 4 ? (g) For what values of x is f (x) 0 ?
5 x 1 Example 3 Consider the function: f (x) x 2 1 (a) Is the point 1, on the graph of f ? 2 (b) If x 2 , what is f (x) ? What point is on the graph of f ? (c) If f (x) 2 , what is x? What point is on the graph of f ? (d) What are the x-intercepts of the graph of f (if any)? What point(s) are on the graph of f?
6 Example 4 The average cost C of manufacturing x computers per day is given by the function 20,000 C(x) 0.56x 2 34.39x 1212.57 x Determine the average cost of manufacturing: (a) 30 computers in a day; ($1351.54) (b) 40 computers in a day; ($1232.97) (c) 50 computers in a day. (1293.07)
7 3.3 Properties of Functions Definition A function f is even if f (x) f (x) . A function f is odd if f (x) f (x) .
Remark (1) A function is even if and only if its graph is symmetric with respect to the y- axis. (2) A function is odd if and only if its graph is symmetric with respect to the origin.
Example 1 Determine whether each graph is the graph of an even function, an odd function, or a function that is neither even nor odd.
Example 2 Determine whether each of the following is even, odd, or neither. Then determine whether the graph is symmetric with respect to the y-axis or with respect to the origin. (a) f (x) x 2 5
(b) g(x) x 3 1
8 (c) h(x) 5x3 x
(d) F(x) x
Definition A function f is increasing on an open interval I if, for any choice of x1 and x2 in I, with x1 < x2 , we have f ( x1 ) < f ( x2 ).
A function f is decreasing on an open interval I if, for any choice of x1 and x2 in I, with x1 < x2 , we have f ( x1 ) > f ( x2 ). A function f is constant on an open interval I if, for any choice of x in I, the values f (x) are equal.
9 Example 3 The graph of a function f is given. Where is f increasing? Where is f decreasing? Where is f constant?
Definition A function f a has a local maximum at c if there is an open interval I containing c so that, for all x ≠c in I, f (x)
Example 4 The graph of a function f is given. (a) At what number(s), if any, does f have a local maximum? (b) What are the local maxima? (c) At what number(s), if any, does f have a local minimum? (d) What are the local minima? (e) List the intervals on which f is increasing. List the intervals on which f is decreasing.
10 Definition If c is in the domain of a function y f (x) , the average rate of change of f from c to x is defined as f (x) f (c) Average rate of change = , x c . x c
Example 5 Find the average rate of change of f (x) 3x 2 : (a) From 1 to 3, (b) From 1 to 5, (c) From 1 to 7
Remark The average rate of change of a function equals the slope of the secant line containing two points on its graph.
Example 6 (a) Find the average rate of change of f (x) 3x 2 2x 3 from -2 to 1. (b) Use this result to find the slope of the secant line containing (2, f (2)) and (1, f (1)) . (c) Find an equation of this secant line.
11 3.4 Library of Functions; Piecewise-defined Functions 1. Library of Functions 1. Constant Function: f (x) b
2. Identity Function: f (x) x
3. Square Function: f (x) x 2
4. Cube Function: f (x) x3
12 5. Square Root Function: f (x) x
6. Cube Root Function: f (x) 3 x
1 7. Reciprocal Function: f (x) x
8. Absolute Value Function: f (x) x
13 9. Greatest integer function: f (x) int (x)
Example If f (x) = int (3x), find: (1) f ( 1.2) (2) f ( 0.1) (3) f ( -1.2) (4) f ( -2.8)
2. Piecewise-defined Functions Definition When functions are defined by more than one equation, they are called piecewise-defined functions.
Example 1 The function f is defined as x 1 if 1 x 1 f (x) 2 if x 1 2 x if x 1
14 (a) Find f (0), f (1) , and f (2) . (b) Determine the domain of f . (c) Graph f . (d) Use the graph to find the range of f .
15 Example 2 In May 2006, CEC supplied electricity to residences for a monthly customer charge of $7.58 plus 8.275 cents per kilowatt hour (kWhr) for the first 400 kWhr supplied in the month and 6.208 cents per kWhr for all usage over 400 kWhr in the month. (a) What is the charge for using 300 kWhr in a month? ($32.41) (b) What is the charge for using 700 kWhr in a month? ($59.30) (c) If C is the monthly charge for x kWhr, express C as a function of x. (0.08275x + 7.58, 0.06208x + 15.848)
16 3.5 Graphing Technique: Transformations
To Graph Draw the Graph of f and: Functional Change to f (x) Vertical shifts
y f (x) k , k 0 Up k units Add k to f (x)
y f (x) k , k 0 Down k units Subtract k from f (x) Horizontal shifts
y f (x h), h 0 Left h units Replace x by x+h
y f (x h), h 0 Right h units Replace x by x-h
Compressing or stretching
y af (x), a 0 Stretch vertically if a 1; Multiply f (x) by a Compress vertically if 0 a 1
y f (ax), a 0 Stretch horizontally if a 1; Replace x by ax Compress horizontally if 0 a 1 Reflection about the x-axis
y f (x) Reflect about x-axis Multiply f (x) by 1 Reflection about the y-axis
y f (x) Reflect about y-axis Replace x by x
Example 1 Use the graph of f (x) x 2 to obtain the graph of g(x) x 2 3 .
17 Example 2 Use the graph of f (x) x 2 to obtain the graph of g(x) x 2 3.
Example 3 Use the graph of f (x) x 2 to obtain the graph of g(x) (x 2) 2 .
Example 4 Use the graph of f (x) x 2 to obtain the graph of g(x) (x 4) 2 .
18 Example 5 Graph the function f (x) (x 3) 2 2 .
Example 6 Use the graph of f (x) x to obtain the graph of g(x) 2 x .
1 Example 7 Use the graph of f (x) x to obtain the graph of g(x) x . 2
19 Example 8 Graph the function f (x) x 2 .
Example 9 Graph the function f (x) x .
Example 10 Find the function that is finally graphed after the following three transformations are applied to the graph of y x . 1) Shift left 2 units 2) Shift up 3 units 3) Reflect about the y-axis.
20 Example 11 Graph the function f (x) x 1 2
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