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

Verified step by step guidance

1. Identify the given function on the graph. Notice the curve is in the fourth quadrant, starting near the y-axis and moving downward as x increases.
2. Recall that the inverse of a function swaps the roles of x and y. This means the inverse function's graph is a reflection of the original graph across the line \(y = x\).
3. To graph the inverse, take several points from the original function, such as \((x, y)\), and switch their coordinates to \((y, x)\). Plot these new points on the coordinate plane.
4. Draw a smooth curve through the new points to form the inverse function. Ensure the shape mirrors the original graph across the line \(y = x\).
5. Label the inverse function clearly and verify that it passes the horizontal line test to confirm it is indeed a function.

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 is also a function, which is guaranteed if the original function is one-to-one.