### About Inverse Functions:

In this section, we test our ability to determine if a function is one-to-one. A function is one-to-one if each y-value of the function corresponds to only one x-value. We can use the horizontal line test to determine if any y-value corresponds to more than one x-value. When a function is one-to-one, we can find the inverse by interchanging the x and y values.

Test Objectives

- Demonstrate the ability to sketch the graph of a function
- Demonstrate the ability to determine if a function is one-to-one
- Demonstrate the ability to find the inverse of a function

#1:

Instructions: Determine if the function is one-to-one.

a) $$f(x)=x^3 + 2$$

b) $$f(x)=2x^2 - 1$$

c) $$h(x)=3|x| - 2$$

Watch the Step by Step Video Solution View the Written Solution

#2:

Instructions: Determine if the function is one-to-one.

a) $$f(x)=-x^2 - 3$$

b) $$f(x)=\sqrt{x}+ 1$$

c) $$f(x)=-4x + 5$$

Watch the Step by Step Video Solution View the Written Solution

#3:

Instructions: Find the inverse.

a) $$h(x)=\frac{- 3}{x - 2}+ 2$$

Watch the Step by Step Video Solution View the Written Solution

#4:

Instructions: Find the inverse.

a) $$f(x)=\frac{1}{x + 3}- 1$$

Watch the Step by Step Video Solution View the Written Solution

#5:

Instructions: Determine if the functions are inverses.

a) $$f(x)=x - 1$$ $$h(x)=x + 1$$

b) $$g(x)=\frac{2}{x}+ 3$$ $$f(x)=\frac{2}{x - 3}$$

c) $$f(x)=1 - \frac{1}{3}x$$ $$g(x)=\frac{-2x + 2}{3}$$

Watch the Step by Step Video Solution View the Written Solution

Written Solutions:

#1:

Solutions:

a) Yes, this function is one-to-one.

b) No, this function is not one-to-one.

c) No, this function is not one-to-one.

Watch the Step by Step Video Solution

#2:

Solutions:

a) No, this function is not one-to-one.

b) Yes, this function is one-to-one.

c) Yes, this function is one-to-one.

Watch the Step by Step Video Solution

#3:

Solutions:

a) $$h^{-1}(x)=\frac{2x - 7}{x - 2}$$

Watch the Step by Step Video Solution

#4:

Solutions:

a) $$f^{-1}(x)=\frac{-3x - 2}{x + 1}$$

Watch the Step by Step Video Solution

#5:

Solutions:

a) f(x) and h(x) are inverses.

b) g(x) and f(x) are inverses.

b) f(x) and g(x) are not inverses.