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

Verified step by step guidance

1. Identify the given function on the graph. Notice the blue line represents the original function, which appears to be a straight line with a negative slope.
2. Recall that the inverse of a function reflects the graph over the line \(y = x\). This means every point \((a, b)\) on the original function will correspond to a point \((b, a)\) on the inverse function.
3. To graph the inverse, start by selecting key points on the original function. For example, pick points where the line crosses grid intersections for easier calculation.
4. Swap the coordinates of these points. For instance, if the original function passes through \((x_1, y_1)\), the inverse will pass through \((y_1, x_1)\).
5. Plot these new points on the graph and draw a line through them. This line will be the inverse function, which should be a reflection of the original line across the line \(y = x\).

Key Concepts

One-to-One Functions

A one-to-one function is a function where each input corresponds to a unique output, and no two different inputs share the same output. This property ensures the function has an inverse because it can be reversed without ambiguity.

Inverse Functions

The inverse of a function reverses the roles of inputs and outputs, swapping x and y values. Graphically, the inverse function is the reflection of the original function across the line y = x.

Graphing and Reflection Across y = x

To graph the inverse of a function, reflect each point of the original function across the line y = x. This means swapping the x- and y-coordinates of each point, which visually demonstrates the inverse relationship.