Five-Minute Check
Then/Now
New Vocabulary
Key Concept: Horizontal Line Test
Example 1: Apply the Horizontal Line Test
Key Concept: Finding an Inverse Function
Example 2: Find Inverse Functions Algebraically
Key Concept: Compositions of Inverse Functions
Example 3: Verify Inverse Functions
Example 4: Find Inverse Functions Graphically
Example 5: Real-World Example: Use an Inverse
Function
Given f(x) = 3x and g(x) = x 2 – 1, find (f ● g)(x)
and its domain.
A.
B.
C.
D.
Given f(x) = 3x and g(x) = x 2 – 1, find
domain.
A.
B.
C.
D.
and its
Given f(x) = 3x and g(x) = x 2 – 1, find [g ○ f](x) and
its domain.
A.
B.
C.
D.
Find two functions f and g such that h(x) = [f ○ g](x).
A.
B.
C.
D.
You found the composition of two functions.
(Lesson 1-6)
• Use the graphs of functions to determine if they
have inverse functions.
• Find inverse functions algebraically and graphically.
• inverse relation
• inverse function
• one-to-one
Apply the Horizontal Line Test
A. Graph the function f(x) = 4x 2 + 4x + 1 using a
graphing calculator, and apply the horizontal line
test to determine whether its inverse function
exists. Write yes or no.
The graph of f(x) = 4x 2 + 4x + 1
shows that it is possible to find a
horizontal line that intersects the
graph of f(x) more than once.
Therefore, you can conclude
that f –1 does not exist.
Answer: no
Apply the Horizontal Line Test
B. Graph the function f(x) = x 5 + x 3 – 1 using a
graphing calculator, and apply the horizontal line
test to determine whether its inverse function
exists. Write yes or no.
The graph of f(x) = x 5 + x 3 – 1
shows that it is not possible to
find a horizontal line that
intersects the graph of f(x)
more than one point.
Therefore, you can conclude
that f –1 exists.
Answer: yes
Graph the function
using a graphing
calculator, and apply the horizontal line test to
determine whether its inverse function exists. Write
yes or no.
C. no
A. yes
B. yes
D. no
Find Inverse Functions Algebraically
A. Determine whether f has an inverse function for
. If it does, find the inverse function
and state any restrictions on its domain.
The graph of f passes the horizontal line test. Therefore,
f is a one-one function and has an inverse function.
From the graph, you can see that f has domain

Now find f –1.
and range

.
Find Inverse Functions Algebraically
Find Inverse Functions Algebraically
Original function
Replace f(x) with y.
Interchange x and y.
2xy – x = y
Multiply each side by 2y – 1. Then
apply the Distributive Property.
2xy – y = x
Isolate the y-terms.
y(2x –1) = x
Factor.
Find Inverse Functions Algebraically
Divide.
Find Inverse Functions Algebraically
From the graph, you can see that f –1 has domain

and range

. The
domain and range of f is equal to the range and
domain of f –1, respectively. Therefore, it is not
necessary to restrict the domain of f –1.
Answer: f –1 exists;
Find Inverse Functions Algebraically
B. Determine whether f has an inverse function for
. If it does, find the inverse function
and state any restrictions on its domain.
The graph of f passes the
horizontal line test. Therefore,
f is a one-one function and
has an inverse function. From
the graph, you can see that f
has domain
and
range
. Now find f –1.
Find Inverse Functions Algebraically
Original function
Replace f(x) with y.
Interchange x and y.
Divide each side by 2.
Square each side.
Find Inverse Functions Algebraically
Add 1 to each side.
Replace y with f –1(x).
From the graph, you can see that f –1 has domain
and range
. By restricting the domain of f –1 to
the range remains
. Only then are the domain and
range of f equal to the range and domain of f –1,
respectively. So,
.
,
Find Inverse Functions Algebraically
Answer: f –1 exists with domain
;
Determine whether f has an inverse function for
. If it does, find the inverse function and
state any restrictions on its domain.
A.
B.
C.
D. f –1(x) does not exist.
Verify Inverse Functions
Show that f[g(x)] = x and g [f(x)] = x.
Verify Inverse Functions
Because f[g(x)] = x and g[f(x)] = x, f(x) and g(x) are
inverse functions. This is supported graphically
because f(x) and g(x) appear to be reflections of each
other in the line y = x.
Verify Inverse Functions
Answer:
Show that f(x) = x 2 – 2, x  0 and
are inverses of each other.
A.
B.
C.
D.
Find Inverse Functions Graphically
Use the graph of relation A to sketch the graph of
its inverse.
Find Inverse Functions Graphically
Graph the line y = x. Locate a few points on the graph
of f(x). Reflect these points in y = x. Then connect
them with a smooth curve that mirrors the curvature of
f(x) in line y = x.
Answer:
Use the graph of the function to
graph its inverse function.
A.
C.
B.
D.
Use an Inverse Function
A. MANUFACTURING The fixed costs for
manufacturing one type of stereo system are
$96,000 with variable cost of $80 per unit. The total
cost f(x) of making x stereos is given by
f(x) = 96,000 + 80x. Explain why the inverse
function f –1(x) exists. Then find f –1(x).
The graph of f(x) = 96,000 +
80x passes the horizontal line
test. Therefore, f(x) is a one-to
one function and has an
inverse function.
Use an Inverse Function
f(x) = 96,000 + 80x
Original function
y = 96,000 + 80x
Replace f(x) with y.
x = 96,000 + 80y
Interchange x and y.
x – 96,000 = 80y
Subtract 96,000 from each
side.
Divide each side by 80.
Replace y with f –1(x).
Use an Inverse Function
Answer: The graph of f(x) passes the horizontal line
test.
Use an Inverse Function
B. MANUFACTURING The fixed costs for
manufacturing one type of stereo system are
$96,000 with variable cost of $80 per unit. The total
cost f(x) of making x stereos is given by
f(x) = 96,000 + 80x. What do f –1(x) and x represent in
the inverse function?
In the inverse function, x represents the total cost and
f –1 (x) represents the number of stereos.
Answer: In the inverse function, x represents the
total cost and f –1(x) represents the number
of stereos.
Use an Inverse Function
C. MANUFACTURING The fixed costs for
manufacturing one type of stereo system are $96,000
with variable cost of $80 per unit. The total cost f(x)
of making x stereos is given by f(x) = 96,000 + 80x.
What restrictions, if any, should be placed on the
domain of f(x) and f –1(x)? Explain.
The function f(x) assumes that the fixed costs are
nonnegative and that the number of stereos is an
integer. Therefore, the domain of f(x) has to be
nonnegative integers. Because the range of f(x) must
equal the domain of f –1(x), the domain of f –1(x) must be
multiples of 80 greater than 96,000.
Use an Inverse Function
Answer: The domain of f(x) has to be nonnegative
integers. The domain of f –1(x) is multiples
of 80 greater than 96,000.
Use an Inverse Function
D. MANUFACTURING The fixed costs for
manufacturing one type of stereo system are
$96,000 with variable cost of $80 per unit. The total
cost f(x) of making x stereos is given by
f(x) = 96,000 + 80x. Find the number of stereos made
if the total cost was $216,000.
Because
the number of stereos made for a total cost of
$216,000 is 1500.
Answer: 1500 stereos
,
EARNINGS Ernesto earns $12 an hour and a
commission of 5% of his total sales as a
salesperson. His total earnings f(x) for a week in
which he worked 40 hours and had a total sales of
$x is given by f(x) = 480 + 0.05x. Explain why the
inverse function f –1(x) exists. Then find f –1(x).
A.
B.
C.
D.