Problem 73 - Precalculus Essentials

Question

Describe how to use the graph of a one-to-one function to draw the graph of its inverse function.

Step-by-Step Solution

Answer

To graph the inverse of a one-to-one function, reflect the graph of the original function over the line \(y = x\). This means for every point (x, y) on the graph of the original function, there will be a point (y, x) on the graph of the inverse function. Also transpose any intercepts and asymptotes.

Step 1: Understanding One-to-One Functions

A function is said to be one-to-one if every x value corresponds to exactly one y value, and every y value corresponds to exactly one x value. This means no two different x-values in the domain map to the same y-value in the range and vice versa.

Step 2: Understanding Inverse Functions

An inverse function is a function which undoes the operation of the original function. In simpler terms, if you have a function that takes x to y, then its inverse function takes y back to x. If \(f(x)\) is the original function then the inverse function is usually written as \(f^{-1}(x)\). For a function to have an inverse, it must be a one-to-one function.

Step 3: Graphing the Inverse

To graph the inverse of a function, reflect the graph of the original function over the line \(y = x\). This is because in the inverse function, the roles of x and y are interchanged, hence the reflection over the line \(y = x\). For every point (a,b) on the graph of \(f(x)\), there will be a point (b,a) on the graph of \(f^{-1}(x)\)

Step 4: Key points

Key points to note in the graphs of function and its inverse are the intercepts and any asymptotes. The x-intercept of the function becomes y-intercept of the inverse function and vice versa. Any horizontal asymptote on the function's graph becomes a vertical asymptote on the inverse's graph and vice versa.