1. Functions and Their Representations
Section 8.1
Functions and Their Representations
Copyright © 2013, 2009, and 2005 Pearson Education, Inc.

2. Objectives Basic Concepts Representations of a Function
Definition of a Function
Identifying a Function
Graphing Calculators (Optional)

3. FUNCTION NOTATION
The notation y = f(x) is called function notation. The input is x, the output is y, and the name of the function is f.
Name
y = f(x)
Output Input

4. The variable y is called the dependent variable and the variable x is called the independent variable. The expression f(4) = 28 is read “f of 4 equals 28” and indicates that f outputs 28 when the input is 4. A function computes exactly one output for each valid input. The letters f, g, and h, are often used to denote names of functions.

5. Representations of a Function
Verbal Representation (Words)
Numerical Representation (Table of Values)
Symbolic Representation (Formula)
Graphical Representation (Graph)
Diagrammatic Representation (Diagram)

6. Example
Evaluate each function f at the given value of x. a. f(x) = 2x – 4, x = –3 b. Solution a. b.

7. Example
Let a function f compute a sales tax of 6% on a purchase of x dollars. Use the given representation to evaluate f(3). Solution a. Verbal Representation Multiply a purchase of x dollars by 0.06 to obtain a sales tax of y dollars. b. Numerical Representation x
f(x)
$1.00
$0.06
$2.00
$0.12
$3.00
$0.18
$4.00
$0.24

8. Example (cont)
c. Symbolic Representation f(x) = 0.06x d. Graphical Representation e. Diagrammatic Representation
1 ●
2 ●
3 ●
4 ●
● 0.06
● 0.12
● 0.18
● 0.24
f(3) = 0.18

9. Definition of a Function
A function receives an input x and produces exactly one output y, which can be expressed as an ordered pair: (x, y). Input Output A relation is a set of ordered pairs, and a function is a special type of relation.

10. Function
A function f is a set of ordered pairs (x, y), where each x-value corresponds to exactly one y-value.
The domain of f is the set of all x-values, and the range of f is the set of all y-values.

11. Example
Use the graph of f to find the function’s domain and range. Solution The arrows at the ends of the graph indicate that the graph extends indefinitely. Thus the domain includes all real numbers. The smallest y-value on the graph is y = −4. Thus the range is y ≥ −4.

13. Example
Use f(x) to find the domain of f. a. f(x) = 3x b. Solution a. Because we can multiply a real number x by 3, f(x) = 3x is defined for all real numbers. Thus the domain of f includes all real numbers. b. Because we cannot divide by 0, the input x = 4 is not valid. The domain of f includes all real numbers except 4, or x ≠ 4.

14. Example Determine whether the table of values represents a function.
f(x) 2 −6 3 4 −1 1
Solution
The table does not represent a function because the input x = 3 produces two outputs; 4 and −1.

15. Vertical Line Test
If every vertical line intersects a graph at no more than one point, then the graph represents a function.

16. Example Determine whether the graphs represent functions. a. b.
Any vertical line will cross the graph at most once. Therefore the graph does represent a function.
The graph does not represent a function because there exist vertical lines that can intersect the graph twice.

17. Section 8.2 Linear Functions
Copyright © 2013, 2009, and 2005 Pearson Education, Inc.

18. Objectives Basic Concepts Representations of Linear Functions
Modeling Data with Linear Functions
The Midpoint Formula (Optional)

19. LINEAR FUNCTION
A function f defined by f(x) = mx + b, where m and b are constants, is a linear function.

20. Example
Determine whether f is a linear function. If f is a linear function, find values for m and b so that f(x) = mx + b. a. f(x) = 6 – 2x b. f(x) = 3x2 – 5 Solution a. Let m = –2 and b = 6. Then f(x) = −2x + 6, and f is a linear function. b. Function f is not linear because its formula contains x2. The formula for a linear function cannot contain an x with an exponent other than 1.

21. Example
Use the table of values to determine whether f(x) could represent a linear function. If f could be linear, write the formula for f in the form f(x) = mx + b. Solution For each unit increase in x, f(x) increases by 7 units so f(x) could be linear with a = 7. Because f(0) = 4, b = 4. thus f(x) = 7x + 4. x 1 2 3
f(x) 4 11 18 25

22. Example
Sketch the graph of f(x) = x – 3 . Use the graph to evaluate f(4). Solution Begin by creating a table.
Plot the points and sketch a line through the points. x y −1 −4 −3 1 −2 2

23. Example (cont)
Sketch the graph of f(x) = x – 3 . Use the graph to evaluate f(4). To evaluate f(4), first find x = 4 on the x-axis. Then find the corresponding y-value. Thus f(4) = 1.

24. MODELING DATA WITH A LINEAR FUNCTION
The formula f(x) = ax + b may be interpreted as follows.
f(x) = mx b
(New amount) = (Change) + (Fixed amount) When x represents time, change equals (rate of change) × (time).
f(x) = m × x b
(Future amount) = (Rate of change) × (Time) + (Initial amount)

25. Example
Suppose that a moving truck costs $0.25 per mile and a fixed rental fee of $20. Find a formula for a linear function that models the rental fees. Solution Total cost is found by multiplying $0.25 (rate per mile) by the number of miles driven x and then adding the fixed rental fee (fixed amount) of $20. Thus f(x) = 0.25x + 20.

26. Example
The temperature of a hot tub is recorded at regular intervals. a. Discuss the temperature of the water during this time interval. The temperature appears to be a constant 102°F. b. Find a formula for a function f that models these data. Because the temperature is constant, the rate of change is 0. Thus f(x) = 0x or f(x) = 102.
Elapsed Time (hours) 1 2 3
Temperature
102°F

27. Example (cont)
The temperature of a hot tub is recorded at regular intervals. c. Sketch a graph of f together with the data.
Elapsed Time (hours) 1 2 3
Temperature
102°F

28. Midpoint Formula in the xy-Plane (Optional)
The midpoint of the line segment with endpoints
(x1, y1) and (x2, y2) in the xy-plane is

29. Example
Find the midpoint of the line segment connecting the points (3, 4) and (5, 3). Solution

30. Compound Inequalities
Section 8.3
Compound Inequalities
Copyright © 2013, 2009, and 2005 Pearson Education, Inc.

31. Objectives Basic Concepts Symbolic Solutions and Number Lines
Numerical and Graphical Solutions
Interval Notation

32. Basic Concepts
A compound inequality consists of two inequalities joined by the words and or or. 2x > –5 and 2x ≤ 8 x + 3 ≥ 4 or x – 2 < –6

33. Example
Determine whether the given x-values are solutions to the compound inequalities. x + 2 < 7 and 2x – 3 > 3 x = 4, –4 Solution x + 2 < 7 and 2x – 3 > 3 Substitute 4 into the given compound inequality < 7 and 2(4) – 3 > 3 6 < 7 and 5 > 3 True and True Both inequalities are true, so 4 is a solution.

34. Example (cont)
Determine whether the given x-values are solutions to the compound inequalities. x + 2 < 7 and 2x – 3 > 3 x = 4, –4 Solution x + 2 < 7 and 2x – 3 > 3 Substitute –4 into the given compound inequality. –4 + 2 < 7 and 2(–4) – 3 > 3 – 2 < 7 and –11 > 3 True and False To be a solution both inequalities must be true, so –4 is not a solution.

35. Symbolic Solutions and Number Lines
We can use a number line to graph solutions to compound inequalities, such as x < 7 and x > –3.
x < 7
x > –3
x < 7 and x > –3
Note: A bracket, either [ or ] or a closed circle is used when an inequality contains ≤ or ≥. A parenthesis, either ( or ), or an open circle is used when an inequality contains < or >.

36. Example
Solve 3x + 6 > 12 and 5 – x < 11 . Graph the solution. Solution 3x + 6 > 12 and 5 – x < 11

37. Example
Solve each inequality. Graph each solution set. Write the solution in set-builder notation. a. b. c. Solution a. b.

38. Example (cont) c.

39. Example
Solve x + 3 < –2 or x + 3 > 2 Solution x + 3 < –2 or x + 3 > 2 x < –5 or x > –1

41. Example
Write each expression in interval notation. a. –3 ≤ x < 7 b. x ≥ 4 c. x < –3 or x ≥ 5 d. {x|x > 0 and x ≤ 5} e. {x|x ≤ 2 or x ≥ 5}

42. Example
Solve 2x + 3 ≤ –3 or 2x + 3 ≥ 5 Solution 2x + 3 ≤ –3 or 2x + 3 ≥ 5 2x ≤ –6 or 2x ≥ 2 x ≤ –3 or x ≥ 1 The solution set may be written as (, 3]  [1, )

43. Other Functions and Their Properties
Section 8.4
Other Functions and Their Properties
Copyright © 2013, 2009, and 2005 Pearson Education, Inc.

44. Objectives Expressing Domain and Range in Interval Notation
Absolute Value Function
Polynomial Functions
Rational Functions (Optional)
Operations on Functions

45. Expressing Domain and Range in Interval Notation
The set of all valid inputs for a function is called the domain, and the set of all outputs from a function is called the range. Rather than writing “the set of all real numbers” for the domain of f, we can use interval notation to express the domain as (−∞, ∞).

46. Example
Write the domain for each function in interval notation. a. f(x) = 3x b. Solution a. The expression 3x is defined for all real numbers x. Thus the domain of f is b. The expression is defined except when x – 4 = 0 or x = 4. Thus the domain of f includes all real numbers except 4 and can be written

47. Absolute Value Function
We can define the absolute value function by f(x) = |x|. To graph y = |x|, we begin by making a table of values. x
|x| –2 2 –1 1

48. Example
Sketch the graph of f(x) = |x – 3|. Write its domain and range in interval notation. Solution Start by making a table of values. x y 3 2 1 4 6
The domain of f is
The range of f is

49. Polynomial Functions
The following expressions are examples of polynomials of one variable. As a result, we say that the following are symbolic representations of polynomial functions of one variable.

50. Example
Determine whether f(x) represents a polynomial function. If possible, identify the type of polynomial function and its degree. a. b. c.
cubic polynomial, of degree 3
not a polynomial function because the exponent on the variable is negative
not a polynomial

51. Example
A graph of is shown. Evaluate f(1) graphically and check your result symbolically. Solution
To calculate f(–1) graphically find –1 on the x-axis and move down until the graph of f is reached. Then move horizontally to the y-axis.
f(1) = –4

52. Example
Evaluate f(x) at the given value of x. Solution

54. Example
Use and to evaluate each of the following. Solution

55. Example (cont)
Use and to evaluate each of the following. Solution

56. Example (cont)
Use and to evaluate each of the following. Solution

57. Absolute Value Equations and Inequalities
Section 8.5
Absolute Value Equations and Inequalities
Copyright © 2013, 2009, and 2005 Pearson Education, Inc.

58. Objectives
Absolute Value Equations
Absolute Value Inequalities

59. Absolute Value Equations
An equation that contains an absolute value is an absolute value equation. Examples: |x| = 2, |2x – 5| = 7, |6 – 2x| – 5 = 2 Consider the equation |x| = 2. This equation has two solutions: 2 and –2 because |2| = 2 and also |2| = 2.

60. Absolute Value Equations

61. Example
Solve each equation. a. |x| = 38 b. |x| = –4 Solution a. |x| = 38 b. |x| = –4

62. Example
Solve |2x – 7| = 5 symbolically. Solution 2x – 7 = 5 or 2x – 7 = –5
Graphical Solution
2x – 7 = 5
or 2x – 7 = –5
2x = 12
or 2x = 2
or x = 1
x = 6
The solutions are 1 and 6.
Numerical Solution x 1 2 3 4 5 6
|2x –7| 7

64. Example
Solve. a. |6 – x| – 3 = 0 b. Solution a. Start by adding 3 to each side.
The solutions are 3 and 9.

65. Example (cont) b. Multiply by 8 to clear fractions.
The solutions are 25/4 and 39/4.

66. Example
Solve. a. |3x – 2| = –7 b. |6 – 3x| = 0 Solution a. |3x – 2| = –7 The absolute value is never negative, there are no solutions.
b. |6 – 3x| = 0
–3x = –6
x = 2

68. Example
Solve |3x| = |2x – 5|. Solution 3x = 2x – 5 or 3x = –(2x – 5) x = –5 or 3x = –2x + 5 The solutions are −5, and 1.
5x = 5
x = 1

70. Example
Solve each absolute value equation and inequality. a. |3 – 4x| = 5 b. |3 – 4x| < 5 c. |3 – 4x| > 5 Solution a. |3 – 4x| = 5 3 – 4x = 5 or 3 – 4x = –5 –4x = 2 or –4x = –8 x = –1/2 or x = 2 b. |3 – 4x| < 5 The solution includes x-values between, but not including –1/2 and 2. {x| –1/2 < x < 2} or (–1/2, 2)

71. Example (cont)
Solve each absolute value equation and inequality. a. |3 – 4x| = 5 b. |3 – 4x| < 5 c. |3 – 4x| > 5 Solution c. |3 – 4x| > 5 The solution includes x-values to the left of x = –1/2 or to the right of x = 2. The solution set is: {x|x < –1/2 or x > 2} or (, –1/2)  (2, )

72. Example
Solve Write the solution set in interval notation. Solution

74. Example
An engineer is designing a circular cover for a container. The diameter d of the cover is to be 4.75 inches and must be accurate within 0.05 inch. Write an absolute value inequality that gives acceptable values for d. Solution The diameter d must satisfy 4.7 ≤ d ≤ 4.8. Subtracting 4.75 from each part gives –0.05 ≤ d – 4.75 ≤ 0.05, which is equivalent to |d – 4.75| ≤ 0.05.

75. Example
Solve if possible. a. b. Solution a. Because the absolute value of an expression cannot be negative, |4 – 3x| is greater than 0 – 1 for every x-value. The solution set is all real numbers.

76. Example
Solve if possible. a. b. Solution a. Because the absolute value is always greater than or equal to 0, no x-values satisfy this inequality. There are no solutions.