Which statement could be used to explain why the function [tex]h(x)=x^3[/tex] has an inverse relation that is also a function?
A. The graph of [tex]h(x)[/tex] passes the vertical line test.
B. The graph of the inverse of [tex]h(x)[/tex] is a vertical line.
C. The graph of the inverse of [tex]h(x)[/tex] passes the horizontal line test.
D. The graph of [tex]h(x)[/tex] passes the horizontal line test.
Answer (2)
The vertical line test checks if a relation is a function.
The horizontal line test checks if the inverse of a function is also a function.
h ( x ) = x 3 has an inverse that is also a function if and only if h ( x ) passes the horizontal line test.
Therefore, the correct answer is: The graph of h ( x ) passes the horizontal line test.
Explanation
Analyzing the Problem We need to determine which statement correctly explains why the function h ( x ) = x 3 has an inverse relation that is also a function. Let's analyze each option.
Evaluating the First Statement The first statement says 'The graph of h ( x ) passes the vertical line test.' The vertical line test determines if a relation is a function. While h ( x ) = x 3 does pass the vertical line test, this doesn't explain why its inverse is also a function.
Evaluating the Second Statement The second statement says 'The graph of the inverse of h ( x ) is a vertical line.' This is incorrect. The inverse of h ( x ) = x 3 is h − 1 ( x ) = 3 x , which is not a vertical line.
Evaluating the Third Statement The third statement says 'The graph of the inverse of h ( x ) passes the horizontal line test.' This is true, and it means that the inverse of the inverse, which is the original function h ( x ) , passes the horizontal line test.
Evaluating the Fourth Statement The fourth statement says 'The graph of h ( x ) passes the horizontal line test.' This is the correct explanation. A function has an inverse that is also a function if and only if it passes the horizontal line test.
Conclusion Therefore, the correct statement is that the graph of h ( x ) passes the horizontal line test.
Examples
Consider a scenario where you want to convert temperatures from Celsius to Fahrenheit and back. The conversion formula is a function, and if its inverse is also a function, it means you can uniquely convert from Fahrenheit back to Celsius without ambiguity. The horizontal line test ensures that for every Fahrenheit temperature, there's only one corresponding Celsius temperature, and vice versa, making the conversion reliable.
The function h ( x ) = x 3 has an inverse that is also a function because it passes the horizontal line test. This test confirms that each output corresponds to exactly one input, making the inverse a function as well. Thus, the correct answer is option D.
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