Name:
Date:
Algebra 2
Inverse Function Formulas & Composition Practice
In the previous lesson, we studied what it means for two functions to be inverses of each other.
Today we answered the following question:
 If you are given a function formula for one function, how do you make a formula for a
second function that will be its inverse
RECAP: How to find an inverse function formula: “change and solve”
Begin with the function in y = ··· notation. (If you have a function name like f(x), change it to y.)
Change every x to a y, and change every y to an x.
Solve the new equation for y.
The result is the inverse function. (Put it back into function notation now, such as f –1(x) = ···.)
Example:
Find the inverse function of f(x) =
y = 23 x  2
Change:
x = 23 y  2
Solve:
x – 2 = 23 y
3
2 (x – 2) = y
3
2 x – 3 = y
Answer:
f –1(x) =
3
2
2
3
x2.
x–3
Classwork/Homework Problems:
1. For each given linear function f(x), find its inverse function. Use the method shown on page 1.
After you have your final answer, rewrite it using the inverse symbol f –1(x) = ···.
a. f(x) = –5x + 4
b. f(x) =
3
4
x6
Name:
Date:
Algebra 2
c. f(x) = – 12 x  10
d. f(x) = 4x 1
2. When a function has the form f(x) = mx + b, it is a linear function whose graph has slope m.
This problem asks you about the slopes of the lines that were in problem 1.
a. Look again at problem 1’s functions f(x) and their inverse functions f –1(x).
What are the slopes of all the functions? Fill in this chart.
slope of f(x)
problem 1a
–5
problem 1b
problem 1c
problem 1d
slope of f –1(x)
b. Look for a pattern in the results. Complete these sentences about what happens when you
make the inverse of a linear function.

Given a linear function f(x), the inverse f –1(x) is also a _____________ function.

The relationship between the slope of f(x) and the slope of f –1(x) is:
_________________________________________________________________.
Name:
Date:
Algebra 2
3. For each function f(x), find the inverse function f –1(x). (Some of these functions aren’t linear, so
don’t expect problem 2’s rule about slopes to always apply here.)
a. f(x) = 20(x – 3) + 100
b. f(x) = 6  x
c. f(x) =
x
4
d. f(x) =
5
x
Name:
Date:
Algebra 2
More Practice: Function Composition
4. Given f ( x)  x 2  x and g ( x)  2 x  3 and h( x)  x find the following:
a. f (g(1))
f. g(g(x))


b. g( f (4))

g. f (h(9))

c. f (g(x))

h. h(h(4))

d. g( f (x))


e. g(g(2))

i. f (h(x))
Name:
Date:
Algebra 2
5. Suppose f x  x 2 and gx   x  6 .
a. Find f g2




c. Suppose hx   ggx  and find h 2


d. Find f ( f (2)) .






f. Find g f g11
e. Find 5 f 3

g. Find g g gg40

b. Find g f 2
Name:
Date:
Algebra 2
ANSWERS:
1. a. f 1 x   
1
1
4
x  4 =   x 
5
5
5
c. f 1 x   2x  10  2 x  20

2. a.
4
x  6 =
3
1
d. f 1 x   x 1 =
4
b. f 1 x  

 1a
problem
slope of f(x)
–5

–1
slope of f (x)
1/5
problem1b
3/4

4/3
4
x8
3
1
1
x
4
4
problem 1c
1/2
2
problem 1d
4
1/4
b. linear, they are reciprocals.
x  100
1
 3=
x 2
20
20
5
d. f –1(x) =
x
3. a. f –1(x) =
b. f –1(x) = 6 – x
c. f –1(x) = 4x

4. Note for problem 4: You don’t need to simplify your formulas, but you can if you want to!
2
a. 30 b. 21 c. f g x   2 x  3  2 x  3  4 x 2  14 x  12
d. g  f x   2 x 2  x  3  2 x 2  2 x  3 e. –1 f. g g x  22x  3  3  4x  9

g. 6 h.

2 i. f hx   x  x
5. a. 16 b. –2 c. –10 d. 16 e. 45 f. 19 g. 16