Name: __________________________________ Block:__________ Date:________________ Paper:______
Inverse Functions
We often know a function that shows how to calculate values of one variable, y, from known
values of another variable, x. But problems in such situations often require discovering the value(s)
of x that will produce specific target values of y.
Suppose you are given the following directions:
 From home, go north on Rt 23 for 5 miles
 Turn east (right) onto Orchard Street
 Go to the 3rd traffic light and turn north (left) onto Baldwin Avenue
 Jekoven’s house is the 5th house on the left.
If you start from Jekoven’s house, write down the directions to get home.
How did you come up with the directions to get home from Jekoven’s?
Suppose you are given the following algorithm:
 Starting with a number, add 5 to it
 Divide the result by 3
 Subtract 4 from that quantity
 Double your result
Write a function f(x), which when given a number x (the original number) will model the operations
given above.
Knowing the final result is 10, work backwards and find the original number. Show your work.
Write a function g(x), which when given a number x (the final result), will model the backward
algorithm that you came up with above.
Fill out the following table:
x
ƒ(x)
g(f(x))
22
1
7
-20
What patterns do you notice in the columns (outputs)?
ƒ(g(ƒ(x)))
Name: __________________________________ Block:__________ Date:________________ Paper:______
The plots of shown below. Notice the symmetry of the two functions when plotted together.
Describe the symmetry in the graph above between the
function and its inverse.
What is the equation for line of symmetry?
In this scenario, f(x) and g(x) are inverses of each other because g(x) will reverse the actions of f(x).
Thus, we could write g(x) as f -1(y) described as “f inverse,” we typically write g=ƒ-1(x). The
exponent “-1” does not mean the reciprocal “one over ƒ(x)” but “the inverse of ƒ(x).”
An inverse function takes and output and gives the corresponding input. Not all functions will have
inverse functions for inverse functions must also pass the vertical line test
If the function ƒ uses x to calculate values of y, the ƒ-1 shows how to use values of y to calculate
corresponding values of x. The challenge is using the rule for ƒ to find the rule for ƒ-1.
The daily profit of the Texas Theatre movie house is a function of the number of paying customers
with rule P(n) = 5n-750
a. What numbers of paying customers will be required for the theater to have:
i. A profit of $100?
ii. A profit of $1250?
iii. A loss of $175?
b. What rule can be used to calculate values for the inverse of the profit function – to find the
number of paying customers required to reach any particular profit target?
Find the inverses of the following functions:
a. If g(x)=3x then g(x)-1 = …
b. If k(x)=(3/5)x+4 then k(x)-1 = …
c. If s(x)=mx+b then s(x)-1 = …
Name: __________________________________ Block:__________ Date:________________ Paper:______
What radical function f -1(y) would undo the actions of the function y=f(x)=x3+2 as shown in the table
and plot below:
x
y=f(x)
f -1(y)
-1
0
1
2
Solution: f -1(y)=___________
Using the table: f -1( f (x))=___________
Line of symmetry: _______________
Study the following strategies for finding the inverse of
Strategy I
If
Then
to see if they are both correct.
Strategy 2
If
Swap the roles of y and x to get
Then
Swap roles of x and y
Then
Then
So,
Hot air balloon flying is a popular recreation in several southwestern states. At some
gatherings of balloonists, there is a competition to hit targets on the ground with bags dropped from
balloons at various altitudes.
The balloons drift toward and over the target at various speeds. To have the best chance that
their drop bags will hit the target balloonists need to know how fast the bags will fall toward the
ground. Principles of physics predict that the velocity in feet per second will be a function of time in
seconds with rule v(t)=32t. The distance fallen in feet will also be a function of time in seconds with
rule d(t) = 16t2.
a. How long will it take the drop bag to reach a velocity of:
i. 48 feet per second?
ii. 100 feet per second?
iii. 150 feet per second?
b. What rule can be used to calculate values for the inverse of the velocity function – to find
the time when any given velocity is reached?
c. How long will it take the drop bag to reach the ground from altitudes of:
i. 144 feet?
ii. 400 feet?
iii. 500 feet?
Name: __________________________________ Block:__________ Date:________________ Paper:______
d. What rule can be used to calculate values of the inverse of the distance function – to find
the time it takes for the drop bad to fall any given distance?
In some situations, you will need to find the inverse of a function that is defined only by table of
values or a graph.
a. Suppose that the function ƒ(x) is as shown in the following table. Make a similar table that shows
assignments bade by ƒ-1(x). Then describe the domain and range if ƒ(x) and ƒ-1(x).
x
-4
-3
-2
-1
0
1
2
3
4
ƒ(x)
2
1
0
-1
-2
-3
-4
-5
-6
-1
ƒ (ƒ(x))
ƒ(x):
Domain:___________
ƒ-1(x): Domain:___________
ƒ(x):
Range: ____________
ƒ-1(x): Range: ____________
b.
i. Plot points made by g-1(x). Then describe the domain and range of g(x) and g-1(x).
g(x):
Domain:___________
g-1(x): Domain:___________
g(x):
Range: ____________
g-1(x): Range: ____________
ii. Draw line segments connecting point (a, b) on the graph of g(x) to the corresponding point
(b, a) on the graph of g-1(x). Then describe the pattern that seems to relate corresponding
points on the graphs of g(x) and g-1(x).
Using what you have learned about the symmetry of ƒ(x) and f -1(x) graph f -1(x) for the following:
Why is y=x the line of symmetry for all of the above graphs?
Name: __________________________________ Block:__________ Date:________________ Paper:______
Additional Practice
Using what you have learned about the symmetry of ƒ(x) and f -1(x) graph f -1(x) for the following:
Why is y=x the line of symmetry for all of the above graphs?
Adapt one strategy or the other to find rules for inverses of the following functions
a.
b.
c.
d.