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

검증된 단계별 안내

Step 1: Understand the concept of inverse functions. A function has an inverse if and only if it is one-to-one, meaning that each input corresponds to exactly one output and vice versa. This can be checked using the Horizontal Line Test.

Step 2: Apply the Horizontal Line Test to each graph in Exercises 96–99. Draw or imagine horizontal lines across the graph. If any horizontal line intersects the graph at more than one point, the function is not one-to-one and does not have an inverse.

Step 3: Identify graphs that pass the Horizontal Line Test. These graphs represent functions that are one-to-one and therefore have inverse functions.

Step 4: For graphs that fail the Horizontal Line Test, note that these functions are not one-to-one and do not have inverse functions. This is because multiple inputs may produce the same output, violating the requirement for an inverse.

Step 5: Summarize your findings by listing which graphs represent functions with inverse functions and which do not, based on the results of the Horizontal Line Test.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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.