Graph the inverse of each one-to-one function.

Verified step by step guidance

1. Identify that the red curve represents a one-to-one function, which means it has an inverse function.
2. Recall that the graph of the inverse function is the reflection of the original function across the line \(y = x\).
3. To graph the inverse, take each point \((a, b)\) on the original function and plot the point \((b, a)\), effectively swapping the \(x\) and \(y\) coordinates.
4. Draw the line \(y = x\) as a reference, which is a diagonal line passing through the origin with a 45-degree angle, to help visualize the reflection.
5. Sketch the inverse function by reflecting the entire red curve across the line \(y = x\), ensuring the shape is mirrored correctly.

Key Concepts

One-to-One Functions

A one-to-one function assigns each input exactly one unique output, and no two different inputs share the same output. This property ensures the function has an inverse, as each output corresponds to only one input.

Inverse Functions

The inverse of a function reverses the roles of inputs and outputs, swapping x and y values. Graphically, the inverse reflects the original function across the line y = x, meaning points (a, b) on the function become (b, a) on its inverse.

Graphing Inverse Functions

To graph an inverse function, reflect the original function's graph over the line y = x. This involves swapping coordinates of key points and ensuring the inverse passes the vertical line test, confirming it is a function.