Holt Algebra 2 7-2 Inverses of Relations and Functions Graph and recognize inverses of relations and...

Holt Algebra 2

7-2 Inverses of Relations and Functions

Graph and recognize inverses of relations and functions.

Find inverses of functions.

Objectives

A relation is a set of ordered pairs. A function is a relation in which each x-value has, at most, one y-value paired with it.

Remember!

Holt Algebra 2


NOTES1. A relation consists of the following points and the

segments drawn between them. Write the ordered pairs for the inverse. And, find the domain and range of the inverse relation:

x 0 3 4 6 9

y 1 2 5 7 8

2. Graph f(x) = 3x – 4. Then write and graph the inverse.

3. A thermometer gives a reading of 25° C. Use the formula C = (F – 32). Write the inverse function and use it to find the equivalent temperature in °F.

59

Holt Algebra 2


You can also find and apply inverses to relations and functions. To graph the inverse relation, you can reflect each point across the line y = x. This is equivalent to switching the x- and y-values in each ordered pair of the relation.

Holt Algebra 2


Graph the relation and connect the points. Then graph the inverse. Identify the domain and range of each relation.

Example 1: Graphing Inverse Relations

x 0 1 5 8

y 2 5 6 9

Graph each ordered pair and connect them.

x 2 5 6 9

y 0 1 5 8

●

●●

●

Switch the x- and y-values in each ordered pair.

Holt Algebra 2


Example 1 Continued

Reflect each point across y = x, and connect them. Make sure the points match those in the table.

Domain:{x|0 ≤ x ≤ 8} Range :{y|2 ≤ x ≤ 9}

Domain:{x|2 ≤ x ≤ 9} Range :{y|0 ≤ x ≤ 8}

•

••

•

••

•

•

Holt Algebra 2


Use inverse operations to write the inverse of f(x) = x – if possible.

Example 2A: Writing Inverses of by Using Inverse Functions

12

is subtracted from the variable, x.12

Add to x to write the inverse.12

f(x) = x – 12

12

f–1(x) = x +

Holt Algebra 2


Use inverse operations to write the inverse of f(x) = .

x3

Example 2B

f–1(x) = 3x

f(x) = The variable x, is divided by 3.

Multiply by 3 to write the inverse.

x3

Holt Algebra 2


Substitute 1 for x.

Check Use the input x = 1 in f(x).

13Substitute for x.

Substitute the result into f–1(x)

= 1

= 13

f–1( ) = 3( )1313

The inverse function does undo the original function.

Example 2B Continued

f(x) = x3

f(1) = 13

f–1(x) = 3x

Holt Algebra 2


You can also find the inverse function by writing the original function with x and y switched and then solving for y.

Holt Algebra 2


Example 3: Writing and Graphing Inverse Functions

Switch x and y.

Solve for y.

Set y = f(x) and graph f.

Graph f(x) = – x – 5. Then write the inverse and graph.

12

12

y = – x – 512

x = – y – 5

x + 5 = – y 12

y = –2x - 10

Holt Algebra 2


Example 3 Continued

f–1(x) = –2x – 10

f

f –1

Graph f(x) = – x – 5. Write the inverse & graph.12

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Anytime you need to undo an operation or work backward from a result to the original input, you can apply inverse functions.

In a real-world situation, don’t switch the variables, because they are named for specific quantities.

Remember!

Holt Algebra 2

7-2 Inverses of Relations and FunctionsExample 4: Retailing Applications

Juan buys a CD online for 20% off the list price. He has to pay $2.50 for shipping. The total charge is $13.70. What is the list price of the CD?

Charge c is a function of list price L.

Step 1 Write an equation for the total charge as a function of the list price.

c = 0.80L + 2.50

Step 2 Find the inverse function that models list price as a function of the change.

c – 2.50 = 0.80L

c – 2.50 = L 0.80

Divide to isolate L.

Holt Algebra 2


Substitute 13.70 for c.

Step 3 Evaluate the inverse function for c = $13.70.

The list price of the CD is $14.

L = 13.70 – 2.50 0.80

Check c = 0.80L + 2.50

= 11.20 + 2.50 = 13.70

Substitute.

= 14

Example 4 Continued

= 0.80(14) + 2.50

Holt Algebra 2


NOTES: Part I

1. A relation consists of the following points and the segments drawn between them. Write the ordered pairs for the inverse. And, find the domain and range of the inverse relation:

D:{x|1 x 8} R:{y|0 y 9}

x 0 3 4 6 9

y 1 2 5 7 8

Holt Algebra 2


Notes: Part II

2. Graph f(x) = 3x – 4. Then write and graph the inverse.

f –1(x) = x + 13

43

f

f –1

Holt Algebra 2


Notes: Part III

3. A thermometer gives a reading of 25° C. Use the formula C = (F – 32). Write the inverse function and use it to find the equivalent temperature in °F.

59

F = C + 32; 77° F95

Holt Algebra 2 7-2 Inverses of Relations and Functions Graph and recognize inverses of relations and...

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