Name: Block: Date: Pre-Calculus 11

Chapter 5A Functions

Lesson #3 Inverse Functions

Investigation

Consider: 𝑓(𝑥) =1

2𝑥 − 4

𝑔(𝑥) = 𝑥2 + 1

1. In the first column of the table below, enter five ordered pairs of f(x) and g(x). In the second

column, interchange the x-coordinates and y-coordinates of the points in the first column.

𝑓(𝑥) =1

2𝑥 − 4 𝑔(𝑥) = 𝑥2 + 1

(x , y) (y , x)

2. Plot the points for the functions f(x) and g(x) for your first column as well as your second

column.

𝑓(𝑥) =1

2𝑥 − 4 𝑔(𝑥) = 𝑥2 + 1

3. What observation can you make about the relationship of the coordinates of your ordered

pairs for your first column and second column of f(x) and g(x)?

(x , y) (y, x)

Name: Block: Date: Pre-Calculus 11

Properties of Inverse Functions

Steps of finding the Inverse of a function

Consider: 𝑓(𝑥) =1

2𝑥 − 4

𝑔(𝑥) = 𝑥2 + 1

Example #1

Consider: 𝑓(𝑥) = 3𝑥 − 2

a) Determine 𝑓−1(𝑥) algebraically

State the domain and range of 𝑓−1(𝑥).

b) Graph both 𝑓(𝑥) and 𝑓−1(𝑥) on the same grid

c) Show that (𝑓(𝑓−1(𝑥)) = 𝑓−1(𝑓(𝑥)) = 𝑥

Name: Block: Date: Pre-Calculus 11

Example #2

Consider: 𝑓(𝑥) = 𝑥2 − 4

a) Determine 𝑓−1(𝑥) algebraically.

State the domain and range of 𝑓−1(𝑥).

b) Graph both 𝑓(𝑥) and 𝑓−1(𝑥) on the same grid

c) Show that (𝑓(𝑓−1(𝑥)) = 𝑓−1(𝑓(𝑥)) = 𝑥

d) Is 𝑓−1(𝑥) a function? If not, describe how the domain of 𝑓(𝑥) could be restricted so that

the inverse of 𝑓−1(𝑥) becomes a function

Name: Block: Date: Pre-Calculus 11

Example #3

Determine whether the functions in each pair are inverses of each other

a) 𝑓(𝑥) = 𝑥 − 4 and 𝑔(𝑥) = 𝑥 + 4

b) 𝑓(𝑥) = 𝑥2 + 1 and 𝑔(𝑥) = √𝑥 + 1

c) 𝑓(𝑥) =(𝑥−2)

2 and 𝑔(𝑥) = 2𝑥 + 2

Example #4

Homework

1. Pg. 268-269 #3, 11, 14, 19, 30, 35, 37, 42, 44, 48 2. Pg. 269-270 #50, 56, 61, 64, 70, 74, 78, 86