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