Problem 50 - College Algebra

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 draw the graph of the inverse of a one-to-one function, reflect the graph of the original function across the line y = x. This involves swapping the x and y values (b, a) of each point on the original graph to plot the inverse graph.

Step 1: Understand the Graph of a One-to-One Function

One-to-one functions are functions that assign unique output values to each input value within their domain. This means that, for a function to be one-to-one, each y-value would correspond to a unique x-value, and all x-values and y-values in the graph would be distinct. Visually, if a horizontal line intersects the graph at more than one point, the function is not one-to-one.

Step 2: Understand the Relationship between a Function and its Inverse

The inverse of a function reverses the role of inputs and outputs. This means that if the original function maps x-values to y-values, the inverse function would map the same y-values back to the respective x-values. Graphically, this implies that the graph of an inverse function is a reflection of the original function’s graph across the line y = x.

Step 3: Draw the Graph of the Inverse Function

To draw the graph of the inverse function, first, draw the line y = x on the same set of axes as the graph of the original function. This line functions as the 'mirror' for the reflection. Then, for each point (a, b) on the original graph, plot a point (b, a) to construct the graph of the inverse function. Essentially, this process involves reflecting each point on the original graph across the line y = x to form the graph of the inverse function.