Question

determine the inverse of h(x), including any restrictions that should be placed on the domains of both h(x) and h⁻¹(x) to ensure that the inverse is a function. use the drop - down menus.

( h(x)=\frac{1}{3}x^2 )

click the arrows to choose an answer from each menu.

if the domain of the original function, h(x), is restricted to choose..., then the inverse of h(x) is the function represented by the equation choose..., where the domain of h⁻¹(x) is restricted to choose...

Explanation:

Step1: Analyze the original function

The function \( h(x)=\frac{1}{3}x^{2} \) is a parabola opening upwards. For it to have an inverse that is a function, we need to restrict its domain to a monotonic interval. The vertex of the parabola is at \( x = 0 \). If we restrict the domain of \( h(x) \) to \( x\geq0 \) (or \( x\leq0 \)), the function will be one - to - one. Let's choose \( x\geq0 \) (the process is similar for \( x\leq0 \)).

Step2: Find the inverse function

To find the inverse, we start by replacing \( h(x) \) with \( y \):
\( y=\frac{1}{3}x^{2} \)
Then, we swap \( x \) and \( y \):
\( x = \frac{1}{3}y^{2} \)
Now, solve for \( y \):
Multiply both sides by 3: \( 3x=y^{2} \)
Take the square root of both sides. Since we restricted the domain of \( h(x) \) to \( x\geq0 \), the range of \( h(x) \) (which will be the domain of \( h^{-1}(x) \)) is \( y\geq0 \), so we take the positive square root: \( y=\sqrt{3x} \) or \( h^{-1}(x)=\sqrt{3x} \) (if we had restricted the domain of \( h(x) \) to \( x\leq0 \), we would have \( h^{-1}(x)=-\sqrt{3x} \))

Step3: Determine the domain of the inverse function

The domain of \( h^{-1}(x) \) is the range of \( h(x) \). For \( h(x)=\frac{1}{3}x^{2} \) with domain \( x\geq0 \), when \( x = 0 \), \( h(0)=0 \), and as \( x \) increases, \( h(x) \) increases. So the range of \( h(x) \) is \( y\geq0 \), which is the domain of \( h^{-1}(x) \).

Answer:

• If the domain of the original function \( h(x) \) is restricted to \( \boldsymbol{x\geq0} \) (or \( x\leq0 \)), then the inverse of \( h(x) \) is the function represented by the equation \( \boldsymbol{h^{-1}(x)=\sqrt{3x}} \) (or \( h^{-1}(x)=-\sqrt{3x} \) if domain of \( h(x) \) is \( x\leq0 \)), where the domain of \( h^{-1}(x) \) is restricted to \( \boldsymbol{x\geq0} \).