Question

Use the graph to determine which statement describes f(x). A. f(x) does not have an inverse function because its graph fails the vertical line test. B. f(x) has an inverse function because its graph passes the horizontal line test. C. f(x) does not have an inverse function because its graph fails the horizontal line test. D. f(x) has an inverse function because its graph passes the vertical line test.

Answer 1

The graph of f(x) is a horizontal line, which fails the horizontal line test, indicating that f(x) does not have an inverse function.

Step-by-step explanation:

To answer the question of whether the function f(x) has an inverse, we must consider the horizontal line test. If any horizontal line intersects the graph of the function at more than one point, then the function does not have a unique inverse because it is not one-to-one. From the description, it seems that the graph of f(x) is a horizontal line, which means it fails the horizontal line test. Therefore, option C is correct: f(x) does not have an inverse function because its graph fails the horizontal line test. It is important to note that the vertical line test is used to determine if a graph represents a function, while the horizontal line test is used to determine if a function has an inverse.

Answer 2

The function described as a horizontal line does not have an inverse function because it fails the horizontal line test. Therefore, the correct statement is 'C: f(x) does not have an inverse function because its graph fails the horizontal line test.'

Step-by-step explanation:

The student's question revolves around determining if a given function f(x) has an inverse function based on its graphical representation and certain tests. Specifically, the question presents four statements and asks to choose the one that correctly describes f(x). Let's analyze the correct statement step by step:

• The vertical line test determines if a graph represents a function. If any vertical line intersects the graph at more than one point, the graph does not represent a function.
• The horizontal line test is used to determine if a function is one-to-one and therefore has an inverse function. If any horizontal line intersects the graph at more than one point, then the function does not have an inverse function.
• Since f(x) is described as a horizontal line, it will fail the horizontal line test because any horizontal line drawn on the graph will coincide with f(x) along its entirety.
• Therefore, the correct statement is C: f(x) does not have an inverse function because its graph fails the horizontal line test.

This logic concludes that the function given does not pass the horizontal line test, indicating that it is not one-to-one and hence does not have an inverse function.