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Precalculus

1.6 Absolute Value Functions

Precalculus1.6 Absolute Value Functions
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  1. Preface
  2. 1 Functions
    1. Introduction to Functions
    2. 1.1 Functions and Function Notation
    3. 1.2 Domain and Range
    4. 1.3 Rates of Change and Behavior of Graphs
    5. 1.4 Composition of Functions
    6. 1.5 Transformation of Functions
    7. 1.6 Absolute Value Functions
    8. 1.7 Inverse Functions
    9. Key Terms
    10. Key Equations
    11. Key Concepts
    12. Review Exercises
    13. Practice Test
  3. 2 Linear Functions
    1. Introduction to Linear Functions
    2. 2.1 Linear Functions
    3. 2.2 Graphs of Linear Functions
    4. 2.3 Modeling with Linear Functions
    5. 2.4 Fitting Linear Models to Data
    6. Key Terms
    7. Key Equations
    8. Key Concepts
    9. Review Exercises
    10. Practice Test
  4. 3 Polynomial and Rational Functions
    1. Introduction to Polynomial and Rational Functions
    2. 3.1 Complex Numbers
    3. 3.2 Quadratic Functions
    4. 3.3 Power Functions and Polynomial Functions
    5. 3.4 Graphs of Polynomial Functions
    6. 3.5 Dividing Polynomials
    7. 3.6 Zeros of Polynomial Functions
    8. 3.7 Rational Functions
    9. 3.8 Inverses and Radical Functions
    10. 3.9 Modeling Using Variation
    11. Key Terms
    12. Key Equations
    13. Key Concepts
    14. Review Exercises
    15. Practice Test
  5. 4 Exponential and Logarithmic Functions
    1. Introduction to Exponential and Logarithmic Functions
    2. 4.1 Exponential Functions
    3. 4.2 Graphs of Exponential Functions
    4. 4.3 Logarithmic Functions
    5. 4.4 Graphs of Logarithmic Functions
    6. 4.5 Logarithmic Properties
    7. 4.6 Exponential and Logarithmic Equations
    8. 4.7 Exponential and Logarithmic Models
    9. 4.8 Fitting Exponential Models to Data
    10. Key Terms
    11. Key Equations
    12. Key Concepts
    13. Review Exercises
    14. Practice Test
  6. 5 Trigonometric Functions
    1. Introduction to Trigonometric Functions
    2. 5.1 Angles
    3. 5.2 Unit Circle: Sine and Cosine Functions
    4. 5.3 The Other Trigonometric Functions
    5. 5.4 Right Triangle Trigonometry
    6. Key Terms
    7. Key Equations
    8. Key Concepts
    9. Review Exercises
    10. Practice Test
  7. 6 Periodic Functions
    1. Introduction to Periodic Functions
    2. 6.1 Graphs of the Sine and Cosine Functions
    3. 6.2 Graphs of the Other Trigonometric Functions
    4. 6.3 Inverse Trigonometric Functions
    5. Key Terms
    6. Key Equations
    7. Key Concepts
    8. Review Exercises
    9. Practice Test
  8. 7 Trigonometric Identities and Equations
    1. Introduction to Trigonometric Identities and Equations
    2. 7.1 Solving Trigonometric Equations with Identities
    3. 7.2 Sum and Difference Identities
    4. 7.3 Double-Angle, Half-Angle, and Reduction Formulas
    5. 7.4 Sum-to-Product and Product-to-Sum Formulas
    6. 7.5 Solving Trigonometric Equations
    7. 7.6 Modeling with Trigonometric Equations
    8. Key Terms
    9. Key Equations
    10. Key Concepts
    11. Review Exercises
    12. Practice Test
  9. 8 Further Applications of Trigonometry
    1. Introduction to Further Applications of Trigonometry
    2. 8.1 Non-right Triangles: Law of Sines
    3. 8.2 Non-right Triangles: Law of Cosines
    4. 8.3 Polar Coordinates
    5. 8.4 Polar Coordinates: Graphs
    6. 8.5 Polar Form of Complex Numbers
    7. 8.6 Parametric Equations
    8. 8.7 Parametric Equations: Graphs
    9. 8.8 Vectors
    10. Key Terms
    11. Key Equations
    12. Key Concepts
    13. Review Exercises
    14. Practice Test
  10. 9 Systems of Equations and Inequalities
    1. Introduction to Systems of Equations and Inequalities
    2. 9.1 Systems of Linear Equations: Two Variables
    3. 9.2 Systems of Linear Equations: Three Variables
    4. 9.3 Systems of Nonlinear Equations and Inequalities: Two Variables
    5. 9.4 Partial Fractions
    6. 9.5 Matrices and Matrix Operations
    7. 9.6 Solving Systems with Gaussian Elimination
    8. 9.7 Solving Systems with Inverses
    9. 9.8 Solving Systems with Cramer's Rule
    10. Key Terms
    11. Key Equations
    12. Key Concepts
    13. Review Exercises
    14. Practice Test
  11. 10 Analytic Geometry
    1. Introduction to Analytic Geometry
    2. 10.1 The Ellipse
    3. 10.2 The Hyperbola
    4. 10.3 The Parabola
    5. 10.4 Rotation of Axes
    6. 10.5 Conic Sections in Polar Coordinates
    7. Key Terms
    8. Key Equations
    9. Key Concepts
    10. Review Exercises
    11. Practice Test
  12. 11 Sequences, Probability and Counting Theory
    1. Introduction to Sequences, Probability and Counting Theory
    2. 11.1 Sequences and Their Notations
    3. 11.2 Arithmetic Sequences
    4. 11.3 Geometric Sequences
    5. 11.4 Series and Their Notations
    6. 11.5 Counting Principles
    7. 11.6 Binomial Theorem
    8. 11.7 Probability
    9. Key Terms
    10. Key Equations
    11. Key Concepts
    12. Review Exercises
    13. Practice Test
  13. 12 Introduction to Calculus
    1. Introduction to Calculus
    2. 12.1 Finding Limits: Numerical and Graphical Approaches
    3. 12.2 Finding Limits: Properties of Limits
    4. 12.3 Continuity
    5. 12.4 Derivatives
    6. Key Terms
    7. Key Equations
    8. Key Concepts
    9. Review Exercises
    10. Practice Test
  14. A | Basic Functions and Identities
  15. Answer Key
    1. Chapter 1
    2. Chapter 2
    3. Chapter 3
    4. Chapter 4
    5. Chapter 5
    6. Chapter 6
    7. Chapter 7
    8. Chapter 8
    9. Chapter 9
    10. Chapter 10
    11. Chapter 11
    12. Chapter 12
  16. Index

Learning Objectives

In this section you will:
  • Graph an absolute value function.
  • Solve an absolute value equation.
  • Solve an absolute value inequality.
The Milky Way.
Figure 1 Distances in deep space can be measured in all directions. As such, it is useful to consider distance in terms of absolute values. (credit: "s58y"/Flickr)

Until the 1920s, the so-called spiral nebulae were believed to be clouds of dust and gas in our own galaxy, some tens of thousands of light years away. Then, astronomer Edwin Hubble proved that these objects are galaxies in their own right, at distances of millions of light years. Today, astronomers can detect galaxies that are billions of light years away. Distances in the universe can be measured in all directions. As such, it is useful to consider distance as an absolute value function. In this section, we will investigate absolute value functions.

Understanding Absolute Value

Recall that in its basic form f(x)=| x |, f(x)=| x |,the absolute value function, is one of our toolkit functions. The absolute value function is commonly thought of as providing the distance the number is from zero on a number line. Algebraically, for whatever the input value is, the output is the value without regard to sign.

Absolute Value Function

The absolute value function can be defined as a piecewise function

f(x)=| x |={ x if x0 x if x<0 f(x)=| x |={ x if x0 x if x<0

Example 1

Determine a Number within a Prescribed Distance

Describe all values x xwithin or including a distance of 4 from the number 5.

Analysis

Note that

4x5 x54 1x x9 4x5 x54 1x x9

So | x5 |4 | x5 |4is equivalent to 1x9. 1x9.

However, mathematicians generally prefer absolute value notation.

Try It #1

Describe all values x xwithin a distance of 3 from the number 2.

Example 2

Resistance of a Resistor

Electrical parts, such as resistors and capacitors, come with specified values of their operating parameters: resistance, capacitance, etc. However, due to imprecision in manufacturing, the actual values of these parameters vary somewhat from piece to piece, even when they are supposed to be the same. The best that manufacturers can do is to try to guarantee that the variations will stay within a specified range, often ±1%,±5%, ±1%,±5%,or ±10%. ±10%.

Suppose we have a resistor rated at 680 ohms, ±5%. ±5%.Use the absolute value function to express the range of possible values of the actual resistance.

Try It #2

Students who score within 20 points of 80 will pass a test. Write this as a distance from 80 using absolute value notation.

Graphing an Absolute Value Function

The most significant feature of the absolute value graph is the corner point at which the graph changes direction. This point is shown at the origin in Figure 3.

Graph of an absolute function
Figure 3

Figure 4 shows the graph of y=2| x3 |+4. y=2| x3 |+4. The graph of y=| x | y=| x | has been shifted right 3 units, vertically stretched by a factor of 2, and shifted up 4 units. This means that the corner point is located at ( 3,4 ) ( 3,4 )for this transformed function.

Graph of the different types of transformations for an absolute function.
Figure 4

Example 3

Writing an Equation for an Absolute Value Function

Write an equation for the function graphed in Figure 5.

Graph of an absolute function.
Figure 5

Analysis

Note that these equations are algebraically equivalent—the stretch for an absolute value function can be written interchangeably as a vertical or horizontal stretch or compression.

Q&A

If we couldn’t observe the stretch of the function from the graphs, could we algebraically determine it?

Yes. If we are unable to determine the stretch based on the width of the graph, we can solve for the stretch factor by putting in a known pair of values for x xand f(x). f(x).

f(x)=a|x3|2 f(x)=a|x3|2

Now substituting in the point (1, 2)

2=a| 13 |2 4=2a a=2 2=a| 13 |2 4=2a a=2
Try It #3

Write the equation for the absolute value function that is horizontally shifted left 2 units, is vertically flipped, and vertically shifted up 3 units.

Q&A

Do the graphs of absolute value functions always intersect the vertical axis? The horizontal axis?

Yes, they always intersect the vertical axis. The graph of an absolute value function will intersect the vertical axis when the input is zero.

No, they do not always intersect the horizontal axis. The graph may or may not intersect the horizontal axis, depending on how the graph has been shifted and reflected. It is possible for the absolute value function to intersect the horizontal axis at zero, one, or two points (see Figure 8).

Graph of the different types of transformations for an absolute function.
Figure 8 (a) The absolute value function does not intersect the horizontal axis. (b) The absolute value function intersects the horizontal axis at one point. (c) The absolute value function intersects the horizontal axis at two points.

Solving an Absolute Value Equation

Now that we can graph an absolute value function, we will learn how to solve an absolute value equation. To solve an equation such as 8=| 2x6 |, 8=| 2x6 |,we notice that the absolute value will be equal to 8 if the quantity inside the absolute value is 8 or -8. This leads to two different equations we can solve independently.

2x6=8 or 2x6=8 2x=14 2x=2 x=7 x=1 2x6=8 or 2x6=8 2x=14 2x=2 x=7 x=1

Knowing how to solve problems involving absolute value functions is useful. For example, we may need to identify numbers or points on a line that are at a specified distance from a given reference point.

An absolute value equation is an equation in which the unknown variable appears in absolute value bars. For example,

| x |=4, | 2x1 |=3 | 5x+2 |4=9 | x |=4, | 2x1 |=3 | 5x+2 |4=9

Solutions to Absolute Value Equations

For real numbers A Aand B, B,an equation of the form | A |=B, | A |=B,with B0, B0,will have solutions when A=B A=Bor A=B. A=B.If B<0, B<0,the equation | A |=B | A |=Bhas no solution.

How To

Given the formula for an absolute value function, find the horizontal intercepts of its graph.

  1. Isolate the absolute value term.
  2. Use | A |=B | A |=Bto write A=B A=Bor −A=B, −A=B, assuming B>0. B>0.
  3. Solve for x. x.

Example 4

Finding the Zeros of an Absolute Value Function

For the function f(x)=| 4x+1 |7 f(x)=| 4x+1 |7, find the values of xx such that  f(x)=0  f(x)=0 .

Try It #4

For the function f(x)=| 2x1 |3, f(x)=| 2x1 |3, find the values of x x such that f(x)=0. f(x)=0.

Q&A

Should we always expect two answers when solving | A |=B? | A |=B?

No. We may find one, two, or even no answers. For example, there is no solution to 2+| 3x5 |=1. 2+| 3x5 |=1.

How To

Given an absolute value equation, solve it.

  1. Isolate the absolute value term.
  2. Use | A |=B | A |=Bto write A=B A=Bor A=−B. A=−B.
  3. Solve for x. x.

Example 5

Solving an Absolute Value Equation

Solve 1=4| x2 |+2. 1=4| x2 |+2.

Q&A

In Example 5, if f(x)=1 f(x)=1and g(x)=4| x2 |+2 g(x)=4| x2 |+2were graphed on the same set of axes, would the graphs intersect?

No. The graphs of f fand g gwould not intersect, as shown in Figure 10. This confirms, graphically, that the equation 1=4| x2 |+2 1=4| x2 |+2 has no solution.

Graph of g(x)=4|x-2|+2 and f(x)=1.
Figure 10
Try It #5

Find where the graph of the function f(x)=| x+2 |+3 f(x)=| x+2 |+3intersects the horizontal and vertical axes.

Solving an Absolute Value Inequality

Absolute value equations may not always involve equalities. Instead, we may need to solve an equation within a range of values. We would use an absolute value inequality to solve such an equation. An absolute value inequality is an equation of the form

|A|<B,|A|B,|A|>B,or |A|B, |A|<B,|A|B,|A|>B,or |A|B,

where an expression A A(and possibly but not usually B B ) depends on a variable x. x.Solving the inequality means finding the set of all x xthat satisfy the inequality. Usually this set will be an interval or the union of two intervals.

There are two basic approaches to solving absolute value inequalities: graphical and algebraic. The advantage of the graphical approach is we can read the solution by interpreting the graphs of two functions. The advantage of the algebraic approach is it yields solutions that may be difficult to read from the graph.

For example, we know that all numbers within 200 units of 0 may be expressed as

| x |<200or200<x<200  | x |<200or200<x<200 

Suppose we want to know all possible returns on an investment if we could earn some amount of money within $200 of $600. We can solve algebraically for the set of values x x such that the distance between x xand 600 is less than 200. We represent the distance between x x and 600 as | x600 |. | x600 |.

|x600|<200    or    200<x600<200   200+600<x600+600<200+600                       400<x<800 |x600|<200    or    200<x600<200   200+600<x600+600<200+600                       400<x<800

This means our returns would be between $400 and $800.

Sometimes an absolute value inequality problem will be presented to us in terms of a shifted and/or stretched or compressed absolute value function, where we must determine for which values of the input the function’s output will be negative or positive.

How To

Given an absolute value inequality of the form | xA |B | xA |B for real numbers a aand b bwhere b bis positive, solve the absolute value inequality algebraically.

  1. Find boundary points by solving | xA |=B. | xA |=B.
  2. Test intervals created by the boundary points to determine where | xA |B. | xA |B.
  3. Write the interval or union of intervals satisfying the inequality in interval, inequality, or set-builder notation.

Example 6

Solving an Absolute Value Inequality

Solve |x5|4. |x5|4.

Analysis

For absolute value inequalities,

|xA|<C, |xA|>C, C<xA<C, xA<C or xA>C. |xA|<C, |xA|>C, C<xA<C, xA<C or xA>C.

The < < or > > symbol may be replaced by  or .  or .

So, for this example, we could use this alternative approach.

|x5|4 4x54 Rewrite by removing the absolute value bars. 4+5x5+54+5 Isolate the x. 1x9 |x5|4 4x54 Rewrite by removing the absolute value bars. 4+5x5+54+5 Isolate the x. 1x9
Try It #6

Solve | x+2 |6. | x+2 |6.

How To

Given an absolute value function, solve for the set of inputs where the output is positive (or negative).

  1. Set the function equal to zero, and solve for the boundary points of the solution set.
  2. Use test points or a graph to determine where the function’s output is positive or negative.

Example 7

Using a Graphical Approach to Solve Absolute Value Inequalities

Given the function f(x)= 1 2 | 4x5 |+3, f(x)= 1 2 | 4x5 |+3, determine the x- x- values for which the function values are negative.

Try It #7

Solve 2| k4 |6. 2| k4 |6.

1.6 Section Exercises

Verbal

1.

How do you solve an absolute value equation?

2.

How can you tell whether an absolute value function has two x-intercepts without graphing the function?

3.

When solving an absolute value function, the isolated absolute value term is equal to a negative number. What does that tell you about the graph of the absolute value function?

4.

How can you use the graph of an absolute value function to determine the x-values for which the function values are negative?

5.

How do you solve an absolute value inequality algebraically?

Algebraic

6.

Describe all numbers x xthat are at a distance of 4 from the number 8. Express this using absolute value notation.

7.

Describe all numbers x xthat are at a distance of 1 2 1 2 from the number −4. Express this using absolute value notation.

8.

Describe the situation in which the distance that point x xis from 10 is at least 15 units. Express this using absolute value notation.

9.

Find all function values f(x) f(x)such that the distance from f(x) f(x)to the value 8 is less than 0.03 units. Express this using absolute value notation.

For the following exercises, solve the equations below and express the answer using set notation.

10.

|x+3|=9 |x+3|=9

11.

|6x|=5 |6x|=5

12.

|5x2|=11 |5x2|=11

13.

|4x2|=11 |4x2|=11

14.

2|4x|=7 2|4x|=7

15.

3|5x|=5 3|5x|=5

16.

3|x+1|4=5 3|x+1|4=5

17.

5| x4 |7=2 5| x4 |7=2

18.

0=| x3 |+2 0=| x3 |+2

19.

2| x3 |+1=2 2| x3 |+1=2

20.

| 3x2 |=7 | 3x2 |=7

21.

| 3x2 |=7 | 3x2 |=7

22.

| 1 2 x5 |=11 | 1 2 x5 |=11

23.

| 1 3 x+5 |=14 | 1 3 x+5 |=14

24.

| 1 3 x+5 |+14=0 | 1 3 x+5 |+14=0

For the following exercises, find the x- and y-intercepts of the graphs of each function.

25.

f(x)=2| x+1 |10 f(x)=2| x+1 |10

26.

f(x)=4| x3 |+4 f(x)=4| x3 |+4

27.

f(x)=3| x2 |1 f(x)=3| x2 |1

28.

f(x)=2| x+1 |+6 f(x)=2| x+1 |+6

For the following exercises, solve each inequality and write the solution in interval notation.

29.

| x2 |>10 | x2 |>10

30.

2| v7 |442 2| v7 |442

31.

| 3x4 |8 | 3x4 |8

32.

| x4 |8 | x4 |8

33.

| 3x5 |13 | 3x5 |13

34.

| 3x5 |13 | 3x5 |13

35.

| 3 4 x5 |7 | 3 4 x5 |7

36.

| 3 4 x5 |+116 | 3 4 x5 |+116

Graphical

For the following exercises, graph the absolute value function. Plot at least five points by hand for each graph.

37.

y=|x1| y=|x1|

38.

y=|x+1| y=|x+1|

39.

y=|x|+1 y=|x|+1

For the following exercises, graph the given functions by hand.

40.

y=| x |2 y=| x |2

41.

y=| x | y=| x |

42.

y=| x |2 y=| x |2

43.

y=| x3 |2 y=| x3 |2

44.

f(x)=|x1|2 f(x)=|x1|2

45.

f(x)=|x+3|+4 f(x)=|x+3|+4

46.

f(x)=2|x+3|+1 f(x)=2|x+3|+1

47.

f(x)=3| x2 |+3 f(x)=3| x2 |+3

48.

f(x)=| 2x4 |3 f(x)=| 2x4 |3

49.

f( x )=| 3x+9 |+2 f( x )=| 3x+9 |+2

50.

f(x)=| x1 |3 f(x)=| x1 |3

51.

f(x)=| x+4 |3 f(x)=| x+4 |3

52.

f(x)= 1 2 | x+4 |3 f(x)= 1 2 | x+4 |3

Technology

53.

Use a graphing utility to graph f(x)=10|x2| f(x)=10|x2| on the viewing window [ 0,4 ]. [ 0,4 ]. Identify the corresponding range. Show the graph.

54.

Use a graphing utility to graph f(x)=100|x|+100 f(x)=100|x|+100on the viewing window [ 5,5 ]. [ 5,5 ].Identify the corresponding range. Show the graph.

For the following exercises, graph each function using a graphing utility. Specify the viewing window.

55.

f(x)=0.1| 0.1(0.2x) |+0.3 f(x)=0.1| 0.1(0.2x) |+0.3

56.

f(x)=4× 10 9 | x(5× 10 9 ) |+2× 10 9 f(x)=4× 10 9 | x(5× 10 9 ) |+2× 10 9

Extensions

For the following exercises, solve the inequality.

57.

|2x 2 3 (x+1)|+3>−1 |2x 2 3 (x+1)|+3>−1

58.

If possible, find all values of a a such that there are no x- x- intercepts for f(x)=2| x+1 |+a. f(x)=2| x+1 |+a.

59.

If possible, find all values of a asuch that there are no y y-intercepts for f(x)=2| x+1 |+a. f(x)=2| x+1 |+a.

Real-World Applications

60.

Cities A and B are on the same east-west line. Assume that city A is located at the origin. If the distance from city A to city B is at least 100 miles and x xrepresents the distance from city B to city A, express this using absolute value notation.

61.

The true proportion p pof people who give a favorable rating to Congress is 8% with a margin of error of 1.5%. Describe this statement using an absolute value equation.

62.

Students who score within 18 points of the number 82 will pass a particular test. Write this statement using absolute value notation and use the variable x xfor the score.

63.

A machinist must produce a bearing that is within 0.01 inches of the correct diameter of 5.0 inches. Using x xas the diameter of the bearing, write this statement using absolute value notation.

64.

The tolerance for a ball bearing is 0.01. If the true diameter of the bearing is to be 2.0 inches and the measured value of the diameter is x xinches, express the tolerance using absolute value notation.

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