Which graphs in Exercises 96–99 represent functions that have inverse functions?

Verified step by step guidance

1. Identify the graphs in Exercises 96-99. Each graph should be analyzed to determine if it represents a function that has an inverse function.
2. Apply the Horizontal Line Test to each graph. This test states that if any horizontal line crosses the graph more than once, then the function does not have an inverse function.
3. For each graph that passes the Horizontal Line Test, conclude that it represents a function with an inverse function. This is because the function is one-to-one, meaning each x-value corresponds to exactly one y-value and vice versa.
4. For graphs that fail the Horizontal Line Test, note that these do not have inverse functions because they are not one-to-one.
5. Summarize which graphs have inverse functions based on the results of the Horizontal Line Test applied to each.

Key Concepts

Function Definition

A function is a relation that assigns exactly one output for each input from its domain. This means that for every x-value, there is a unique y-value. Understanding the definition of a function is crucial for determining whether a graph represents a function and whether it can have an inverse.

Horizontal Line Test

The horizontal line test is a method used to determine if a function has an inverse that is also a function. If any horizontal line intersects the graph of the function more than once, the function does not have an inverse that is a function. This test is essential for analyzing the graphs in the given exercises.

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, meaning it passes the horizontal line test, ensuring that each output corresponds to a unique input.