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

Verified step by step guidance

1. <insert step 1> Identify the graphs that represent functions. A graph represents a function if any vertical line drawn through the graph intersects it at most once. This is known as the Vertical Line Test.>
2. <insert step 2> Determine if the function is one-to-one. A function is one-to-one if any horizontal line drawn through the graph intersects it at most once. This is known as the Horizontal Line Test.>
3. <insert step 3> If a function passes both the Vertical Line Test and the Horizontal Line Test, it is a one-to-one function and has an inverse function.>
4. <insert step 4> For each graph, apply the Vertical Line Test and the Horizontal Line Test to check if the function is one-to-one.>
5. <insert step 5> Conclude which graphs represent functions that have inverse functions based on the tests applied.>

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.