Question

Describe the relationship between the graph of a function and the graph of its inverse function.

Short answer

The relationship between the graph of a function and the graph of its inverse function is shown by the graph of the inverse function being a reflection of the graph of the original function across the line y=x. If the graph of function f crosses the point (a, b), then the graph of the inverse function f^-1 crosses the point (b, a) and vice versa.

Step-by-step solution

Understand the inverse function

The inverse function is a concept in mathematics that 'reverses' the inputs and outputs of the initial function. If the original function turns x into y, the inverse function turns y back into x. This is akin to something like undoing the original function.

Reflect across the line y=x

Graphically, the inverse function appears as a reflection of the original function about the line y=x. This means that every point (x, y) on the graph of the original function corresponds to point (y, x) on the graph of the inverse function.

Confirm the relationship

Through the observation from steps 1 and 2, we can confirm the relationship between a function and its inverse. If the function f takes x to y, then the inverse function (usually denoted as f^-1) takes y back to x. This is shown graphically as the reflection of the function across the line y=x. If the graph of function f crosses the point (a, b), then the graph of the inverse function f^-1 crosses the point (b, a), and vice versa.