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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
7-2 Inverses of Relations and Functions
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
7-2 Inverses of Relations and Functions
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
7-2 Inverses of Relations and Functions
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
7-2 Inverses of Relations and Functions
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}
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•
••
•
•
Holt Algebra 2
7-2 Inverses of Relations and Functions
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
7-2 Inverses of Relations and Functions
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
7-2 Inverses of Relations and Functions
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
7-2 Inverses of Relations and Functions
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
7-2 Inverses of Relations and Functions
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
7-2 Inverses of Relations and Functions
Example 3 Continued
f–1(x) = –2x – 10
f
f –1
Graph f(x) = – x – 5. Write the inverse & graph.12
Holt Algebra 2
7-2 Inverses of Relations and Functions
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
7-2 Inverses of Relations and Functions
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
7-2 Inverses of Relations and Functions
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
7-2 Inverses of Relations and Functions
Notes: Part II
2. Graph f(x) = 3x – 4. Then write and graph the inverse.
f –1(x) = x + 13
43
f
f –1