5.5 Building Inverses of Functions
Today’s goals are:
(1)
(2)
(3)
(4)
(5)
Define the inverse relation of a function
Given several points in a function, find inverse points.
See relation-inverse symmetry across the line y = x
Find the composition of functions with their inverses
Apply inverses in real-world situations
-----------------------------------------------------------------------------------------------------------------------------------------Introduction:
Gloria and Keith are sharing their graphs for the same set of data.
“I know my graph is right!” exclaims Gloria. “I’ve check and rechecked it. Your must be wrong, Keith.”
Keith disagrees. “I’ve entered these data into my calculator too, and I made sure I entered the correct numbers.”
The graphs are pictured below. Can you explain what is happening?
Vocabulary
Inverse
Recall* Relation vs. Function
Inverse of a Relation
One-to-one function
Investigation: The Inverse
In this investigation you will use graphs, tables, and equations to explore the inverses of several functions. You may
need a separate sheet of paper to do your work.
Step 1
Graph the equation f(x) = 6 + 3x on your calculator. Complete the table for this function.
X
Y
-1
0
1
2
3
Step 2
Because the inverse is obtained by switching the independent and dependent variables, you can find
five points on the inverse of function f by swapping the x- and y-coordinates in the table. Complete the table for the
inverse.
X
Y
-1
0
1
2
3
Step 3
Graph the five points you found in Step 2 by creating a scatter plot. Describe the graph and write an
equation for it. Graph your equation and verify that it passes through the points in the table from Step 2.
Step 4
Repeat Steps 1-3 for each of these functions. You may need to write more than one equation to
describe the inverse.
i.
g(x) =
x 1  3
ii.
h(x) = (x – 2)2 – 5
Step 5
Study the graphs of functions and their inverses that you made. What observations can you make about
the graphs of a function and its inverse?
Step 6
You create the inverse by switching the x- and y-values of points. How can you apply this idea to find
the equation of the inverse from the original function? Verify that your method works by using it to find the equations
for the inverses of functions f, g, and h.
Example A: A 589 mi flight from Washington, D.C., to Chicago took 118 min. A flight of 1452 mi from Washington, D.C.,
to Denver took 222 min. Model this relationship both as (time, distance) data and as (distance, time) data. If a flight
from Washington, D.C., to Seattle takes 323 min, what is the distance traveled? If the distance between Washington,
D.C., and Miami is 910 mi, how long will it take to fly from one of these two cities to the other?
Example B: Find the composition of this function with its inverse.
f(x) = 4 – 3x
Practice 5.5
1.
A function f(x) contains the points (-2, -3), (0,-1), (2,2), and (4,6). Give the points known to be in the inverse of
f(x).
2.
a.
Given g(t) = 5 + 2t, find each value.
g(2)
b. g-1(9)
3.
Which graph below represents the inverse of the relation shown in the graph at right? Explain how you know.
4.
Match each function with its inverse.
a.
y = 6 – 2x
e. y 
1
( x  6)
2
b. y  2 
f. y 
c. g-1(20)
6
x
2
x6
c. y = -6(x – 2)
g. y  2 
1
x
6
d. y 
6
x2
h. y  6 
2
x
5. Given f(x) = 4 + (x – 2)3/5
a. Solve for x when f(x) = 12
b. Find f-1(x) symbolically
c. How are solving for x and finding an inverse alike? How are they different?
6. Consider the graph of the piecewise function f shown at right.
a. Find f(-3), f(-1), f(0), and f(2).
b. Name four points on the graph of the inverse of f(x)
c. Draw the graph of the inverse. Is the inverse of function? Explain.
Write each function using f(x) notation, then find its inverse. If the inverse if a function write it using f-1(x)
notation.
a. y = 2x – 3
b. 3x + 2y = 4
c. x2 + 2y = 3
7.
Name: ______________________
Date:_______________
Homework 5.4 Building Inverses of Functions
1.
Each of the functions below has an inverse that is also a function.; Find four points on the graph of each
function f, using the given values of x. Use these points to find four points on the graph of f-1.
a. f(x) = 3x – 4; x = -2, 0, 4/3, 4
b. f(x) = x3 – 2; x = -3, -1, 2, 5
2.
For each function below, determine whether or not the inverse of the function is a function. Find the equation
of the inverse and graph both equations on the same axes.
a. y = -2x + 5
b. y = |x|
c. y = x2 - 4
3.
Balloons and Laughs Inc. is a small company that entertains at children’s birthday parties. B & L uses a
complicated formula to calculate its prices, taking into account all of its costs. The price equations if p(x) =
3
(8 x  3) 2  25 , where x is the number of person-hours supplied for the party at a price of p(x). For example
if x = 4, four clowns will come for one hour, two clowns will come for two hours, or one clown will come for four
hours.
a. What is the price if two clowns come to a party for 90 minutes?
b.
Many customers want to know what they can get for a particular amount of money. Rewrite the price
equations for B & L so that the company can input the amount of money a customer wants to spend and the
output will be the number of person-hours he or she will get for the money. Call the new function p-1(x).
c. B&L’s Ultimate Party costs $125. How many person-hours do you get at an Ultimate Party?
4.
Rewrite the expression 1252/3 in as many different ways as you can.
5.
Find the exponential function that contains the points (2, 12.6) and (5, 42.525)
6.
Solve by rewrite with the same base.
x
3
=9
x
1
c. 2 =  
4
x
x-3
a.
4 =8
7.
Write the equation of a parabola with vertex (3,2) passing through the point (4,5).
8.
Solve this system of equations.
 x  3 y  z  4

2 x  x  y
2.2 y  2.2 z  2.2

b. 3
4x+1
Name: __________________________________
Exit Ticket 5.5
Given the equation:
y = (x – 3)2 + 1
Select all that apply:






The vertex is at (3, -1)
The graph has been reflected over the x-axis
The graph has been dilated by 3
The graph shifted the parent function up 1
The graph is a parabola
The graph of the inverse is a parabola

The inverse of the function is y = 

The inverse of the function is y = ( x  1) 2  3



The graph goes through the point (5, 5)
The vertex of the inverse is (1, 3)
The equation is a power function
x 1  3
Name: __________________________________
Exit Ticket 5.5
Given the equation:
y = (x – 3)2 + 1
Select all that apply:






The vertex is at (3, -1)
The graph has been reflected over the x-axis
The graph has been dilated by 3
The graph shifted the parent function up 1
The graph is a parabola
The graph of the inverse is a parabola

The inverse of the function is y = 

The inverse of the function is y = ( x  1) 2  3



The graph goes through the point (5, 5)
The vertex of the inverse is (1, 3)
The equation is a power function
x 1  3
Do Now:
Let f(x) = 2x – 8 and g(x) =
Find f(5)
Find x when f(x) = 2.
Find g(2)
Find x when g(x) = 5
Find f(g(x))
Find g(f(x))
1
x4
2