Each of Exercises 19-24 gives a formula for a function y=f(x) and shows the graphs of f and f^-1

step 1 In this problem, we're given the function f of x equals x to the two thirds. We're given the gra

Each of Exercises 19-24 gives a formula for a function...

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Key Concepts

Inverse Function

An inverse function essentially reverses the operation of the original function. If a function f maps an input x to an output y, then its inverse f?¹ maps y back to x. This idea is fundamental in many areas of mathematics, as it allows one to undo the effect of a function under the assumption that the function is one-to-one (bijective).

Finding an Inverse by Swapping Variables

To determine an inverse function algebraically, one typically starts with the equation y = f(x) and then swaps the roles of x and y. After interchanging the variables, one solves the resulting equation for y, which then becomes the inverse function f?¹(x). This method is central to understanding and finding the inverse of a given function.

One-to-One Function Property

A function must be one-to-one (or injective) to have an inverse function, meaning that every output is produced by exactly one input. This property ensures that the process of finding an inverse by swapping variables is valid and that the inverse function will pass the vertical line test. It guarantees that no two distinct x-values map to the same y-value.

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