Final answer:
The function h(x)=x^3 has an inverse that is also a function because its graph passes the vertical line test, meaning that each x-value corresponds to exactly one y-value.
Step-by-step explanation:
To explain why the function h(x)=x^3 has an inverse relationship that is also a function, we should look at the options provided:
- (a) The graph of h(x) is a straight line.
- (b) The graph of h(x) passes the vertical line test.
- (c) The graph of h(x) is a perfect circle.
- (d) The graph of h(x) has multiple y-values for a single x-value.
The correct answer is (b) The graph of h(x) passes the vertical line test. A function's graph must pass the vertical line test, meaning that a vertical line drawn at any x-coordinate on the graph only touches the graph at one point. Since h(x)=x^3 is a cubic function, its graph is a curve that passes the vertical line test and does not repeat any y-value for a given x-value. This means that each x-value has a unique y-value, allowing an inverse function to exist and be a function as well because it will pass the horizontal line test, which is necessary for a function to have an inverse that is also a function.