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
Inverse Functions Practice Test:

#1:

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

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

f(x)=x^3 + 2 graphed

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

f(x) = 2x^2 - 1 graphed

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

h(x) = 3|x| - 2 graphed

#2:

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

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

f(x) = -x^2 - 3 graphed

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

f(x) = sqrt(x) + 1 graphed

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

f(x) = -4x + 5 graphed

#3:

Instructions: Find the inverse.

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


#4:

Instructions: Find the inverse.

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


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


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.


#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.


#3:

Solutions:

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


#4:

Solutions:

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


#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.