Answer :
To determine why the function [tex]\( h(x) = x^3 \)[/tex] has an inverse relation that is also a function, we need to delve into a rigorous understanding of function inverses and relevant graph tests.
### Understanding an Inverse Function
For a function [tex]\( h(x) \)[/tex] to have an inverse [tex]\( h^{-1}(x) \)[/tex] that is also a function, [tex]\( h(x) \)[/tex] must be a one-to-one function. This means that for every element [tex]\( y \)[/tex] in the range of the function, there is exactly one [tex]\( x \)[/tex] in the domain of the function such that [tex]\( h(x) = y \)[/tex].
### Graphical Tests
Various graphical tests can help determine whether a function and its inverse meet the necessary conditions:
1. Vertical Line Test:
- This test determines whether a graph represents a function. A graph passes the vertical line test if no vertical line intersects the graph at more than one point.
- For [tex]\( h(x) = x^3 \)[/tex], the graph passes the vertical line test, confirming that [tex]\( h(x) \)[/tex] is indeed a function. However, this does not directly tell us anything about the inverse.
2. Horizontal Line Test:
- This test determines whether a function is one-to-one. A graph passes the horizontal line test if no horizontal line intersects the graph at more than one point.
- For [tex]\( h(x) = x^3 \)[/tex], the graph passes the horizontal line test since any horizontal line intersects the cubic curve at exactly one point. This confirms that [tex]\( h(x) = x^3 \)[/tex] is one-to-one and thus has an inverse that is also a function.
3. Graph of the Inverse's Tests:
- Considering the graph of the inverse function itself can also provide insights. Specially, we might look at whether the graph of the inverse passes certain tests that confirm it behaves like a function itself.
- Specifically, if the graph of the inverse of [tex]\( h(x) \)[/tex] passes the horizontal line test, it assures us that the inverse relation is indeed a function.
### Step-by-Step Reasoning
- The function [tex]\( h(x) = x^3 \)[/tex] passes the vertical line test, confirming that [tex]\( h(x) \)[/tex] is a function.
- Next, the function [tex]\( h(x) = x^3 \)[/tex] passes the horizontal line test, confirming that [tex]\( h(x) \)[/tex] is one-to-one.
- Because [tex]\( h(x) \)[/tex] passes the horizontal line test, its inverse relation [tex]\( h^{-1}(x) \)[/tex] exists and is also a function.
Therefore, the correct statement to explain why the function [tex]\( h(x) = x^3 \)[/tex] has an inverse relation that is also a function is:
The graph of the inverse of [tex]\( h(x) \)[/tex] passes the horizontal line test.
### Understanding an Inverse Function
For a function [tex]\( h(x) \)[/tex] to have an inverse [tex]\( h^{-1}(x) \)[/tex] that is also a function, [tex]\( h(x) \)[/tex] must be a one-to-one function. This means that for every element [tex]\( y \)[/tex] in the range of the function, there is exactly one [tex]\( x \)[/tex] in the domain of the function such that [tex]\( h(x) = y \)[/tex].
### Graphical Tests
Various graphical tests can help determine whether a function and its inverse meet the necessary conditions:
1. Vertical Line Test:
- This test determines whether a graph represents a function. A graph passes the vertical line test if no vertical line intersects the graph at more than one point.
- For [tex]\( h(x) = x^3 \)[/tex], the graph passes the vertical line test, confirming that [tex]\( h(x) \)[/tex] is indeed a function. However, this does not directly tell us anything about the inverse.
2. Horizontal Line Test:
- This test determines whether a function is one-to-one. A graph passes the horizontal line test if no horizontal line intersects the graph at more than one point.
- For [tex]\( h(x) = x^3 \)[/tex], the graph passes the horizontal line test since any horizontal line intersects the cubic curve at exactly one point. This confirms that [tex]\( h(x) = x^3 \)[/tex] is one-to-one and thus has an inverse that is also a function.
3. Graph of the Inverse's Tests:
- Considering the graph of the inverse function itself can also provide insights. Specially, we might look at whether the graph of the inverse passes certain tests that confirm it behaves like a function itself.
- Specifically, if the graph of the inverse of [tex]\( h(x) \)[/tex] passes the horizontal line test, it assures us that the inverse relation is indeed a function.
### Step-by-Step Reasoning
- The function [tex]\( h(x) = x^3 \)[/tex] passes the vertical line test, confirming that [tex]\( h(x) \)[/tex] is a function.
- Next, the function [tex]\( h(x) = x^3 \)[/tex] passes the horizontal line test, confirming that [tex]\( h(x) \)[/tex] is one-to-one.
- Because [tex]\( h(x) \)[/tex] passes the horizontal line test, its inverse relation [tex]\( h^{-1}(x) \)[/tex] exists and is also a function.
Therefore, the correct statement to explain why the function [tex]\( h(x) = x^3 \)[/tex] has an inverse relation that is also a function is:
The graph of the inverse of [tex]\( h(x) \)[/tex] passes the horizontal line test.
