Which graphs in Exercises 29–34 represent functions that have inverse functions?

Verified step by step guidance

1. Recall that a function has an inverse function if and only if it is one-to-one, meaning each output corresponds to exactly one input.
2. Use the Horizontal Line Test on each graph: if any horizontal line intersects the graph more than once, the graph does not represent a one-to-one function and therefore does not have an inverse function.
3. For each graph, imagine drawing horizontal lines across the entire domain and check if any line crosses the graph more than once.
4. If a graph passes the Horizontal Line Test (no horizontal line intersects it more than once), then the function represented by that graph has an inverse function.
5. Identify and list the graphs from Exercises 29–34 that pass this test to determine which represent functions with inverse functions.

Key Concepts

Definition of a Function

A function is a relation where each input corresponds to exactly one output. Understanding this ensures that when analyzing graphs, each x-value has only one y-value, which is essential before considering inverses.

Inverse Functions

An inverse function reverses the roles of inputs and outputs of the original function. For a function to have an inverse, it must be one-to-one, meaning no two different inputs share the same output.

Horizontal Line Test

The horizontal line test determines if a function is one-to-one by checking if any horizontal line intersects the graph more than once. Passing this test indicates the function has an inverse function.