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 how many times each line intersects the graph.
4. If every horizontal line intersects the graph at most once, then the function is one-to-one and has an inverse function.
5. Summarize which graphs pass the Horizontal Line Test and thus 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.

One-to-One Functions

A function has an inverse only if it is one-to-one, meaning each output corresponds to exactly one input. This property ensures the inverse relation is also a function, which is critical when identifying invertible graphs.

Horizontal Line Test

The horizontal line test is a graphical method to determine if a function is one-to-one. If any horizontal line intersects the graph more than once, the function fails the test and does not have an inverse function.