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 function if it passes the horizontal line test.
The horizontal line test checks if any horizontal line intersects the graph of the function at most once.
The function h ( x ) = x 3 passes the horizontal line test.
Therefore, the graph of h ( x ) passes the horizontal line test, which explains why its inverse is also a function. The graph of h ( x ) passes the horizontal line test.
Explanation
Understanding the Problem The question asks us to identify the statement that explains why the function h ( x ) = x 3 has an inverse relation that is also a function. We need to recall the conditions under which a function has an inverse that is also a function.
Recalling the Horizontal Line Test A function has an inverse that is also a function if and only if it is a one-to-one function (injective). A one-to-one function means that for every y value, there is only one x value that corresponds to it. Graphically, this is determined by the horizontal line test. If any horizontal line intersects the graph of the function at most once, then the function is one-to-one and has an inverse function.
Understanding the Vertical Line Test The vertical line test, on the other hand, is used to determine 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.
Applying the Horizontal Line Test to h(x) The function h ( x ) = x 3 is a one-to-one function. Its graph passes the horizontal line test. Therefore, 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 can represent the volume in cubic meters, and its inverse would give you the side length of a cube with a given volume. Since each volume corresponds to a unique side length, the inverse is also a function. This concept is useful in various real-world applications, such as calculating dimensions, converting units, and understanding relationships between quantities.
The function h ( x ) = x 3 has an inverse that is also a function because it passes the horizontal line test, meaning no horizontal line intersects its graph more than once. This confirms that h ( x ) is a one-to-one function. Thus, the correct answer is that the graph of h ( x ) passes the horizontal line test.
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