Inverse Functions Worksheet A 2.8 AP Precalculus

Inverse Functions Worksheet A 2.8 AP Precalculus

Inverse functions are crucial in precalculus, particularly for AP students. This worksheet focuses on identifying and finding inverse functions, including methods for determining if a function is invertible. It includes exercises on sketching graphs of inverse functions and understanding the relationship between a function and its inverse. Designed for AP Precalculus students, this resource aids in mastering key concepts necessary for success in the course and on the exam.

Key Points

  • Explains the concept of inverse functions and their properties in precalculus.
  • Includes exercises for determining the invertibility of various functions.
  • Provides methods for sketching the graphs of inverse functions.
  • Covers the relationship between a function and its inverse through composition.
  • Offers practice problems to reinforce understanding of inverse functions.
, ,
  • newtopiccyclegrowin
68
/ 5
© The Algebros from FlippedMath.com
A function, 𝑓, has an inverse function, or is
__________
, if each output value of 𝑓 is mapped from a
unique input value.
𝑓
󰇛
𝑥
󰇜
󰇛𝑥2󰇜
3
Don’t get confused by the notation for inverse compared to the reciprocal.
An inverse function is a reverse mapping of the function. That is, if 𝑓
󰇛
𝑎
󰇜
𝑏, then 𝑓

󰇛
𝑏
󰇜
𝑎.
Another way of thinking of this is if a function has the coordinate pair
󰇛
𝑎, 𝑏
󰇜
, then the inverse function
has the coordinate pair
󰇛
𝑏, 𝑎
󰇜
If the function is increasing or decreasing only, then it is invertible. If the graph “turns” (has a max or
min), then it is no longer invertible because there will be output values that are the same for different
input values. Think of this as a “horizontal line test”. The Vertical Line Test checks to see if a graph is
a function. The Horizontal Line Test checks to see if a graph’s inverse is a function.
Are the following functions invertible? Sketch the graph of the inverse.
1.
2.
The inverse of the graph of the function 𝑓
󰇛
𝑥
󰇜
can be found by reversing the roles of the 𝑥- and 𝑦-axes.
This means we can reflect the graph of 𝑓 over the line 𝑦𝑥 to get the graph of the inverse.
The domain and range of a function and its inverse are swapped.
What is the minimum value of 𝑓

󰇛
𝑥
󰇜
?
What is the maximum value of 𝑓

󰇛
𝑥
󰇜
?
2.8InverseFunctions
𝒙
𝒇
𝟏
󰇛
𝒙
󰇜
𝒙
𝒇
󰇛
𝒙
󰇜
𝑓
󰇛𝑥󰇜
𝑔󰇛𝑥󰇜
APPrecalc
Writeyourquestions
andthoughtshere!
2.8Notes
      




y
𝑓
󰇛𝑥󰇜
© The Algebros from FlippedMath.com
One method of finding the inverse function is to reverse the roles of
𝑥
and
𝑦
in the equation, then solve
for
𝑦
.
3. Find the inverse function of
𝑓
󰇛
𝑥
󰇜
󰇛𝑥2󰇜
3
.

The domain of a function can be restricted to make the function invertible.
4. Find the inverse function of
𝑓
󰇛
𝑥
󰇜
𝑥
2
Restrict the domain to the left side.
What is the domain and range of the inverse function?
Find the inverse function along with the domain and range of the inverse.
5.
𝑓
󰇛
𝑥
󰇜
√
𝑥3
2
Domain of
𝑓

.
Range of
𝑓

.
6.
𝑓
󰇛
𝑥
󰇜

Domain of
𝑓

.
Range of
𝑓

.
7.
𝑓
󰇛
𝑥
󰇜


Domain of
𝑓

.
Range of
𝑓

.
Composition of 𝒇 and 𝒇
𝟏
The composition of a function, 𝑓, and its inverse function 𝑓

, is the identity function.
𝑓𝑓

󰇛
𝑥
󰇜
𝑓

𝑓
󰇛
𝑥
󰇜
𝑥
8. Are
𝑓
󰇛
𝑥
󰇜

and
𝑔
󰇛
𝑥
󰇜
3
inverses?
Writeyourquestions
andthoughtshere!
© The Algebros from FlippedMath.com
2.8InverseFunctions
APPrecalculus
Find the inverse of each function and list the domain and range of
𝒇
𝟏
󰇛
𝒙
󰇜
.
1.
𝑓
󰇛
𝑥
󰇜
󰇛𝑥3󰇜
4
Domain of 𝑓

󰇛
𝑥
󰇜
:
Range of 𝑓

󰇛
𝑥
󰇜
:
2.
𝑓
󰇛
𝑥
󰇜
𝑥6
Domain of 𝑓

󰇛
𝑥
󰇜
:
Range of 𝑓

󰇛
𝑥
󰇜
:
3.
𝑓
󰇛
𝑥
󰇜
󰇛
𝑥1
󰇜
2 for 𝑥1
Domain of 𝑓

󰇛
𝑥
󰇜
:
Range of 𝑓

󰇛
𝑥
󰇜
:
4.
𝑓
󰇛
𝑥
󰇜
√
𝑥2 3
Domain of 𝑓

󰇛
𝑥
󰇜
:
Range of 𝑓

󰇛
𝑥
󰇜
:
5.
𝑓
󰇛
𝑥
󰇜
󰇛𝑥2󰇜
5 for 𝑥2
Domain of 𝑓

󰇛
𝑥
󰇜
:
Range of 𝑓

󰇛
𝑥
󰇜
:
6.
𝑓
󰇛
𝑥
󰇜

Domain of 𝑓

󰇛
𝑥
󰇜
:
Range of 𝑓

󰇛
𝑥
󰇜
:
7.
𝑓
󰇛
𝑥
󰇜

󰇛
𝑥4
󰇜
1 for 𝑥4
Domain of 𝑓

󰇛
𝑥
󰇜
:
Range of 𝑓

󰇛
𝑥
󰇜
:
8.
𝑓
󰇛
𝑥
󰇜

√
𝑥1 3
Domain of 𝑓

󰇛
𝑥
󰇜
:
Range of 𝑓

󰇛
𝑥
󰇜
:
9.
𝑓
󰇛
𝑥
󰇜
2𝑥5
3𝑥4
Domain of 𝑓

󰇛
𝑥
󰇜
:
Range of 𝑓

󰇛
𝑥
󰇜
:
2.8Practice
/ 5
End of Document
68
You May Also Like

FAQs of Inverse Functions Worksheet A 2.8 AP Precalculus

What are the properties of inverse functions?
Inverse functions reverse the mapping of the original function, meaning if f(a) = b, then f⁻¹(b) = a. A function must be one-to-one to have an inverse, ensuring each output corresponds to a unique input. The horizontal line test is often used to determine if a function is invertible; if any horizontal line intersects the graph more than once, the function is not one-to-one. Understanding these properties is essential for solving problems involving inverse functions.
How do you find the inverse of a function?
To find the inverse of a function, start by replacing f(x) with y. Then, swap the x and y variables and solve for y. This process effectively reverses the roles of the input and output. After isolating y, rename it as f⁻¹(x) to denote the inverse function. It's important to check the domain and range of the original function to ensure the inverse is valid.
What is the significance of the horizontal line test?
The horizontal line test is a graphical method used to determine if a function is one-to-one and thus invertible. If any horizontal line intersects the graph of the function more than once, it indicates that the function has the same output for different inputs, meaning it is not one-to-one. Consequently, such a function does not have an inverse. This test is essential for precalculus students to understand when studying inverse functions.
What types of functions are typically invertible?
Typically, functions that are strictly increasing or strictly decreasing are invertible. These functions do not repeat output values for different input values, satisfying the one-to-one condition necessary for an inverse. Examples include linear functions with a non-zero slope and certain polynomial functions. Understanding which functions are invertible helps students apply the correct methods for finding inverses in precalculus.
How does the composition of a function and its inverse work?
The composition of a function f and its inverse f⁻¹ results in the identity function. This means that f(f⁻¹(x)) = x and f⁻¹(f(x)) = x for all x in the domain of f and f⁻¹. This property illustrates the fundamental relationship between a function and its inverse, reinforcing the concept that they essentially undo each other. Mastery of this concept is crucial for students in AP Precalculus.

Related of Inverse Functions Worksheet A 2.8 AP Precalculus