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

Verified step by step guidance

1. Identify the given function on the graph. The red curve represents the function.
2. To find the inverse, reflect the graph of the function over the line y = x. This means swapping the x and y coordinates of each point on the original graph.
3. Plot the reflected points on the graph. For example, if the original function passes through (1, 2), the inverse will pass through (2, 1).
4. Draw a smooth curve through the reflected points to represent the inverse function.
5. Verify that the new graph is a reflection of the original graph over the line y = x.

Key Concepts

One-to-One Functions

A one-to-one function is a type of function where each output value is uniquely paired with one input value. This means that no two different inputs produce the same output. To determine if a function is one-to-one, the horizontal line test can be applied: if any horizontal line intersects the graph of the function more than once, the function is not one-to-one.

Inverse Functions

An inverse function essentially reverses the effect of the original function. If a function f takes an input x to an output y, then its inverse f⁻¹ takes y back to x. For a function to have an inverse, it must be one-to-one, ensuring that each output corresponds to exactly one input, allowing for a unique reversal.

Graphing Inverses

When graphing the inverse of a function, the graph of the inverse can be obtained by reflecting the original function across the line y = x. This reflection indicates that the roles of the x and y coordinates are swapped. Understanding this geometric relationship is crucial for accurately sketching the inverse function based on the original graph.