Inverses and Radical Functions: Fresh Take

  • Learn how to find the inverse (or “reverse”) of a polynomial function when it’s possible
  • Figure out how to limit the domain of a polynomial function so you can find its inverse
  • Use radical functions to solve real-world problems

Radical Functions

The Main Idea

  1. Radical Functions:
    • General form: [latex]f(x) = \sqrt[n]{g(x)}[/latex]
    • [latex]n[/latex] is a positive integer (root degree)
    • [latex]g(x)[/latex] is any function of [latex]x[/latex]
  2. Inverse Functions:
    • Notation: [latex]f^{-1}(x)[/latex]
    • Property: [latex]f^{-1}(f(x)) = x[/latex] and [latex]f(f^{-1}(x)) = x[/latex]
    • Graphical relationship: Symmetric about [latex]y = x[/latex]
  3. One-to-One Functions:
    • Definition: Each output corresponds to a unique input
    • Test: Horizontal line test
    • Importance: Only one-to-one functions have inverses that are functions

Types of Radical Functions

  1. Square Root Function: [latex]f(x) = \sqrt{x}[/latex]
  2. Cube Root Function: [latex]f(x) = \sqrt[3]{x}[/latex]
  3. Higher-Order Root Functions: [latex]f(x) = \sqrt[n]{x}[/latex], where [latex]n \geq 3[/latex]

Steps to Finding Inverse Functions:

  1. Verify the function is one-to-one
  2. Replace [latex]f(x)[/latex] with [latex]y[/latex]
  3. Interchange [latex]x[/latex] and [latex]y[/latex]
  4. Solve for [latex]y[/latex]
  5. Rename the function [latex]f^{-1}(x)[/latex]
Show that [latex]f\left(x\right)=\dfrac{1}{x+1}[/latex] and [latex]{f}^{-1}\left(x\right)=\dfrac{1}{x}-1[/latex] are inverses, for [latex]x\ne 0,-1[/latex] .

Find the inverse function of [latex]f\left(x\right)=\sqrt[3]{x+4}[/latex].

Domains of Radical Functions

The Main Idea

  • The domain of a radical function depends on the index [latex]n[/latex] of the root:
    • For even roots (e.g., square roots), the expression inside the radical must be non-negative.
    • For odd roots (e.g., cube roots), the expression inside the radical can be any real number.
  • To find the domain of a radical function:
    • Set up an inequality ensuring the radicand is non-negative (for even roots).
    • Identify critical points where the expression could change sign.
    • Test intervals and determine where the inequality holds.
  • Not all functions are one-to-one. To find an inverse of a non-one-to-one function:
    • Restrict the domain to make the function one-to-one.
    • Find the inverse on this restricted domain.
  • Steps to find the inverse of a function:
    • Replace [latex]f(x)[/latex] with [latex]y[/latex].
    • Interchange [latex]x[/latex] and [latex]y[/latex].
    • Solve for [latex]y[/latex].
    • Rename the function [latex]f^{-1}(x)[/latex].
    • Ensure the domain of [latex]f^{-1}(x)[/latex] corresponds to the range of [latex]f(x)[/latex].
  • For radical functions, when finding the inverse:
    • Determine the range of the original function.
    • This range becomes the domain restriction for the inverse function.
  • Graphically, inverse functions are reflections of each other over the line [latex]y = x[/latex].
  • The domain of [latex]f(x)[/latex] becomes the range of [latex]f^{-1}(x)[/latex], and vice versa.
  • When graphing radical functions and their inverses:
    • Points of intersection will always lie on the line [latex]y = x[/latex].
    • If [latex](a, b)[/latex] is on the graph of [latex]f(x)[/latex], then [latex](b, a)[/latex] is on the graph of [latex]f^{-1}(x)[/latex].
Find the inverse of the function [latex]f\left(x\right)={x}^{2}+1[/latex], on the domain [latex]x\ge 0[/latex].

Watch the following video to see more examples of how to restrict the domain of a quadratic function to find it’s inverse.

You can view the transcript for “Ex: Restrict the Domain to Make a Function 1 to 1, Then Find the Inverse” here (opens in new window).