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)
A function has an inverse that is also a function if it is one-to-one.
A function is one-to-one if it passes the horizontal line test.
The function h ( x ) = x 3 passes the horizontal line test.
Therefore, the graph of h ( x ) passes the horizontal line test, explaining why its inverse is also a function. T h e g r a p h o f h ( x ) p a sses t h e h or i zo n t a l l in e t es t .
Explanation
Understanding the Problem The question asks us to identify the correct statement that explains why the function h ( x ) = x 3 has an inverse relation that is also a function. We need to consider the properties of functions and their inverses, particularly the horizontal and vertical line tests.
One-to-one Functions A function has an inverse that is also a function if and only if the original function is one-to-one (injective). A one-to-one function means that for every y value, there is only one corresponding x value.
Horizontal Line Test The horizontal line test is a visual way to determine if a function is one-to-one. If any horizontal line intersects the graph of the function at most once, then the function is one-to-one.
Vertical Line Test The vertical line test determines if a relation is a function. If any vertical line intersects the graph of a relation at most once, then the relation is a function. This test is used to check if a graph represents a function, not whether its inverse is a function.
Applying the Horizontal Line Test to h(x) Since we want to know why the inverse of h ( x ) = x 3 is also a function, we need to determine if h ( x ) is one-to-one. This is determined by the horizontal line test. The graph of h ( x ) = x 3 passes the horizontal line test, meaning that any horizontal line will intersect the graph at most once. Therefore, h ( x ) is one-to-one, and its inverse is also a function.
Conclusion Therefore, the correct statement is: The graph of h ( x ) passes the horizontal line test.
Examples
Consider a scenario where you want to convert cubic meters to liters. The function h ( x ) = x 3 could represent a simplified model relating a certain input to its cube. If this relationship is one-to-one (passes the horizontal line test), it means that for every volume in cubic meters, there's a unique volume in liters, and vice versa. This ensures a consistent and reversible conversion process, which is crucial in many scientific and engineering applications where accurate unit conversions are necessary.
The function h ( x ) = x 3 is one-to-one because it passes the horizontal line test, which allows its inverse to also be a function. Therefore, the correct statement is the graph of h ( x ) passes the horizontal line test. The selected option is C.
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