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Lesson 7: The inverse of a function

Topic. This lesson covers Session 7: The inverse of a function

Learning Outcomes.

  • Identify one-to-one functions and understand the connection to inverse functions.
  • Form connections between the definition of inverse functions, the notation of inverse functions, and the application of inverse functions.
  • Find the inverse of a function graphically and algebraically.

WeBWorK. There is one WeBWorK assignment on today’s material:

  1. Functions – Inverse Functions

Additional Video Resources.

Question of the Day: What is the opposite of x^3?

Topic 1

Definition. A function f is called one-to-one (or injective), if two different inputs x_{1} \neq x_{2} always have different outputs f\left(x_{1}\right) \neq f\left(x_{2}\right).

Example. Consider the functions f(x)=x^2 and g(x)=x^3, shown in the diagram below. Are either of these functions one-to-one?

Graphs of f(x)=x^2 and g(x)=x^3

Observation (Horizontal Line Test). A function is one-to-one exactly when every horizontal line intersects the graph of the function at most once.

A function is one-to-one when each output is determined by exactly one input. Therefore we can construct a new function, called the inverse function, where we reverse the roles of inputs and outputs.

Definition 7.5. Let f be a function with domain D_{f} and range R_{f}, and assume that f is one-to-one. The inverse of f is a function f^{-1} so that

    \[f(x)=y \text{ means precisely that } f^{-1}(y)=x\]

Diagram showing the relationship between inputs and outputs of f and f inverse.

Example. Find the inverse of each function:

  • f(x)=2x-7
  • g(x)=\sqrt{x+2}
  • h(x)=\frac{x+1}{x+2}

VIDEO: Finding the Inverse of a Function

Video by Irania Vazquez

Exit Question

Find the inverse of f(x)=x^3 + 7

Answer

f^{-1}(x)=\sqrt[3]{x-7}

Good job! You are now ready to practice on your own – give the WeBWorK assignment a try. If you get stuck, try using the “Ask for Help” button to ask a question on the Q&A site.