7-2 Inverses of Relations and Functions
Objectives
Graph and recognize inverses of
relations and functions.
Find inverses of functions.
Remember!
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.
Holt Algebra 2
7-2 Inverses of Relations and Functions
NOTES
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:
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 = 5 (F – 32). Write the inverse
9
function and use it to find the equivalent
temperature in °F.
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 yvalues in each ordered pair of the relation.
Holt Algebra 2
7-2 Inverses of Relations and Functions
Example 1: Graphing Inverse Relations
Graph the relation and
connect the points. Then
graph the inverse. Identify
the domain and range of each
relation.
Graph each ordered pair and
connect them.
Switch the x- and y-values in
each ordered pair.
x
y
Holt Algebra 2
2
0
5
1
6
5
9
8
x
0 1 5
8
y
2 5 6
9
●
●
●
●
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}
Holt Algebra 2
7-2 Inverses of Relations and Functions
Example 2A: Writing Inverses of by Using Inverse
Functions
Use inverse operations to write the inverse of
f(x) = x – 1 if possible.
2
f(x) = x – 1
1 is subtracted from the variable, x.
2
f–1(x) = x + 1
Add 21 to x to write the inverse.
2
2
Holt Algebra 2
7-2 Inverses of Relations and Functions
Example 2B
Use inverse operations to write the inverse of
f(x) = x .
3
f(x) =
x
3
f–1(x) = 3x
Holt Algebra 2
The variable x, is divided by 3.
Multiply by 3 to write the inverse.
7-2 Inverses of Relations and Functions
Example 2B Continued
Check Use the input x = 1 in f(x).
f(x) = x3
f(1) = 1
3
Substitute 1 for x.
= 1
3
Substitute the result into f–1(x)
f–1(x) = 3x
1
1
Substitute 31 for x.
f–1( 3 ) = 3( 3 )
=1
The inverse function does undo the original function. 
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
1
Graph f(x) = – 2
x – 5. Then write the inverse
and graph.
1
y=– 2
x–5
1
x=– 2
y–5
1
x+5=–2
y
y = –2x - 10
Holt Algebra 2
Set y = f(x) and graph f.
Switch x and y.
Solve for y.
7-2 Inverses of Relations and Functions
Example 3 Continued
Graph f(x) = – 1 x – 5. Write the inverse & graph.
2
f–1(x) = –2x – 10
f
f –1
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.
Remember!
In a real-world situation, don’t switch the
variables, because they are named for specific
quantities.
Holt Algebra 2
7-2 Inverses of Relations and Functions
Example 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?
Step 1 Write an equation for the total charge as a
function of the list price.
c = 0.80L + 2.50
Charge c is a function of list price L.
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
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Divide to isolate L.
7-2 Inverses of Relations and Functions
Example 4 Continued
Step 3 Evaluate the inverse function for c = $13.70.
L = 13.70 – 2.50
0.80
Substitute 13.70 for c.
= 14
The list price of the CD is $14.
Check c = 0.80L + 2.50
= 0.80(14) + 2.50
= 11.20 + 2.50
= 13.70 
Holt Algebra 2
Substitute.
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:
x
0
3
4
6
9
y
1
2
5
7
8
D:{x|1  x  8}
Holt Algebra 2
R:{y|0  y  9}
7-2 Inverses of Relations and Functions
Notes: Part II
2. Graph f(x) = 3x – 4. Then write and graph
the inverse.
f
f –1
1
4
f –1(x) = 3 x + 3
Holt Algebra 2
7-2 Inverses of Relations and Functions
Notes: Part III
3. A thermometer gives a reading of 25° C. Use the
formula C = 5 (F – 32). Write the inverse
9
function and use it to find the equivalent
temperature in °F.
9
F= 5
C + 32; 77° F
Holt Algebra 2