Is The Inverse Of $h(x)=x^3$ A Function?

Hey guys! Let's dive into a super interesting math question today: Why does the function $h(x)=x^3$ have an inverse relation that is also a function? This might sound a bit technical, but trust me, once we break it down, it's going to make a lot of sense. We're going to explore the properties of the function $h(x)=x^3$ and its inverse, and figure out what makes its inverse a function. Get ready to flex those math muscles!

Understanding Inverse Relations

First off, what exactly is an inverse relation? Think of it like this: if a function takes an input and gives you an output, its inverse relation does the opposite. It takes the output of the original function and gives you back the original input. For our function $h(x)=x^3$, if we plug in a number, say 2, we get $h(2) = 2^3 = 8$. The inverse relation would then take 8 and give us back 2. Mathematically, if we have a function $y = f(x)$, its inverse relation is found by swapping $x$ and $y$, so we get $x = f(y)$. To express this as a function of $x$, we'd solve for $y$. For $h(x)=x^3$, the equation is $y = x^3$. Swapping $x$ and $y$ gives us $x = y^3$. To find the inverse function, we solve for $y$ by taking the cube root of both sides: $y = \sqrt[3]{x}$. So, the inverse relation of $h(x)=x^3$ is $h^{-1}(x) = \sqrt[3]{x}$. Now, the big question is, why is this inverse relation $h^{-1}(x) = \sqrt[3]{x}$ also a function? We've got a few options to consider, and we'll explore each one.

Option A: The Vertical Line Test

Let's talk about Option A: The graph of $h(x)$ passes the vertical line test. What does the vertical line test even mean, you ask? Well, visually, it's a way to check if a graph represents a function. You imagine drawing vertical lines across the graph. If any vertical line crosses the graph more than once, then it's not a function. If every vertical line you draw crosses the graph at most once, then it is a function. This test works because a function, by definition, can only have one output for each input. So, if a vertical line hits the graph more than once, it means there's an $x$-value that corresponds to multiple $y$-values, which is a no-go for functions.

Now, let's apply this to our original function, $h(x)=x^3$. The graph of $y=x^3$ is a smooth, S-shaped curve that goes up from left to right. If you draw any vertical line across this graph, it will always intersect the curve at exactly one point. This is because for every $x$ you choose, there's only one possible value for $x^3$. So, yes, the graph of $h(x)=x^3$ does pass the vertical line test. This tells us that $h(x)=x^3$ itself is indeed a function. But does this directly tell us if its inverse is a function? That's where things get a little more nuanced, and we need to consider how the inverse relates to the original function's properties.

Option B: The Graph of the Inverse is a Vertical Line

Moving on to Option B: The graph of the inverse of $h(x)$ is a vertical line. Okay, let's think about this. What does a vertical line graph look like? A vertical line graph is something like $x = c$, where $c$ is a constant. For example, $x = 5$. If you try to apply the vertical line test to a vertical line graph, it fails miserably! In fact, every vertical line you draw (except for the line itself, which is technically $x=c$ and passes it infinitely many times) will intersect the graph at infinitely many points. This means a vertical line graph (other than a single point) cannot represent a function. Since we're trying to determine if the inverse of $h(x)=x^3$ is a function, and we know the inverse is $h^{-1}(x) = \sqrt[3]{x}$, let's think about its graph. The graph of $y = \sqrt[3]{x}$ is definitely not a vertical line. It's a curve that looks somewhat like a sideways 'S'. Therefore, this statement, "The graph of the inverse of $h(x)$ is a vertical line," is false, and it certainly doesn't explain why the inverse is a function. In fact, if it were a vertical line, the inverse would not be a function!

Option C: The Graph of the Inverse Passes the Vertical Line Test

Now, let's consider Option C: The graph of the inverse of $h(x)$ passes the vertical line test. This one sounds promising, right? Remember, the vertical line test is the gold standard for determining if a graph represents a function. If the graph of the inverse relation passes this test, it means that for every $x$-value in the domain of the inverse, there is only one corresponding $y$-value. This is the very definition of a function!

So, let's look at the inverse of $h(x)=x^3$, which we found to be $h^{-1}(x) = \sqrt[3]{x}$. What does the graph of $y = \sqrt[3]{x}$ look like? It's a curve that passes through the origin (0,0). If $x$ is positive, $y$ is positive, and as $x$ gets larger, $y$ also gets larger, but not as quickly as $x^3$. If $x$ is negative, $y$ is also negative. Imagine plotting points: $(-8, -2)$, $(-1, -1)$, $(0, 0)$, $(1, 1)$, $(8, 2)$.

Now, try drawing vertical lines across this graph. Does any vertical line hit the graph more than once? Nope! Each vertical line will intersect the graph of $y = \sqrt[3]{x}$ at exactly one point. This is because for any given $x$-value, there is only one real cube root. For instance, the cube root of 8 is uniquely 2, and the cube root of -8 is uniquely -2. Since the graph of $h^{-1}(x) = \sqrt[3]{x}$ passes the vertical line test, this statement "The graph of the inverse of $h(x)$ passes the vertical line test" is true and is the correct explanation for why the inverse relation is also a function.

Option D: The Graph of $h(x)$ Passes the Horizontal Line Test

Let's chat about Option D: The graph of $h(x)$ passes the horizontal line test. This is a really important one when we're talking about inverse functions, guys. While the vertical line test checks if a relation is a function, the horizontal line test checks if a function has an inverse that is also a function. How does it work? You imagine drawing horizontal lines across the graph of a function. If any horizontal line crosses the graph more than once, it means that there are multiple $x$-values that produce the same $y$-value. If this happens, the inverse relation will not be a function, because when you swap $x$ and $y$, that single $y$-value from the original function will correspond to multiple $x$-values in the inverse, violating the vertical line test for the inverse.

However, if every horizontal line you draw crosses the graph at most once, then the function does have an inverse that is also a function. Why? Because this condition ensures that each output $y$ from the original function comes from a unique input $x$. When you flip $x$ and $y$ to get the inverse, each input $x$ in the inverse will uniquely map to an output $y$. This is exactly what the vertical line test for the inverse requires!

Now, let's apply this to $h(x)=x^3$. The graph of $y=x^3$ is that S-shaped curve. If you draw any horizontal line across this graph, it will intersect the curve at exactly one point. For example, the line $y=8$ only intersects $y=x^3$ at $x=2$. The line $y=-27$ only intersects $y=x^3$ at $x=-3$. Because $h(x)=x^3$ passes the horizontal line test, this confirms that its inverse relation, $h^{-1}(x)=\sqrt[3]{x}$, must be a function. So, this statement, "The graph of $h(x)$ passes the horizontal line test," is also true and provides a fundamental reason why the inverse is a function. It's directly linked to the inverse passing the vertical line test.

Connecting the Dots: Horizontal Line Test vs. Vertical Line Test for Inverse

So, we've established that both Option C and Option D are true statements about $h(x)=x^3$ and its inverse. But the question asks why the inverse relation is a function. Let's clarify the roles:

• Vertical Line Test (for the inverse): This directly tells us if the inverse relation is a function. If the graph of the inverse passes the VLT, it is a function. This is Option C.
• Horizontal Line Test (for the original function): This tells us if the original function's inverse will be a function. If the original function passes the HLT, its inverse will be a function. This is Option D.

Essentially, Option D (HLT for $h(x)$) is the precondition that guarantees Option C (VLT for $h^{-1}(x)$) will be true. They are deeply connected!

Think about it: if $h(x)$ fails the HLT, it means there's a horizontal line that hits the graph of $h(x)$ more than once. Let's say this line is $y=k$, and it hits $h(x)$ at $x_1$ and $x_2$ (where $x_1 \neq x_2$). So, $h(x_1)=k$ and $h(x_2)=k$. Now, when we find the inverse, we swap $x$ and $y$. The points $(x_1, k)$ and $(x_2, k)$ become $(k, x_1)$ and $(k, x_2)$ on the graph of the inverse. This means the input $k$ for the inverse has two different outputs, $x_1$ and $x_2$. If you draw a vertical line at $x=k$ on the graph of the inverse, it will hit two points. Thus, the inverse fails the vertical line test and is not a function.

Conversely, if $h(x)$ passes the HLT, every horizontal line hits it at most once. This means each output $y$ comes from a unique input $x$. When we swap $x$ and $y$ for the inverse, each input $x$ in the inverse will lead to a unique output $y$, meaning the inverse passes the vertical line test and is a function.

Final Verdict

So, to directly explain why the inverse relation $h^{-1}(x) = \sqrt[3]{x}$ is a function, we need to confirm that its graph satisfies the definition of a function. The most direct way to do this visually is to see if it passes the vertical line test. And indeed, the graph of $y = \sqrt[3]{x}$ does pass the vertical line test. Therefore, Option C: The graph of the inverse of $h(x)$ passes the vertical line test is the most direct and correct explanation.

While Option D is also true and explains why the inverse is a function (because the original function has this property), Option C is the property that the inverse itself must possess to be a function. The question asks to explain why the inverse relation is also a function, and the vertical line test is the definition of a function in graphical terms. Keep these tests straight, guys – they're super handy!