Question

select the correct answer from each drop - down menu. a tennis ball is dropped from the roof of a building at a height of 25 feet. function h models the height of the tennis ball above the ground, h(x), in feet, x seconds after it is dropped: h(x)=-16x² + 25. complete the statements about the inverse function, h⁻¹. the inverse function will be a function and have limits on the domain. function h⁻¹ will give h⁻¹(x), in terms of x,

Explanation:

Step1: Find the inverse of the function

Let $y = h(x)=-16x^{2}+25$. Swap $x$ and $y$: $x=-16y^{2}+25$. Then solve for $y$:

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Since $x$ represents time and is non - negative in the context of the ball being dropped ($h(x)$ is a real - valued function for the physical situation where the ball is dropped from a height and falls downwards), and the height $h(x)\geq0$, for the inverse function we consider the positive square root. The original function $h(x)=-16x^{2}+25$ is a parabola opening downwards. The domain of $h(x)$ is $x\geq0$ (time starts at $0$ when the ball is dropped) and the range is $h(x)\in[0,25]$. For the inverse function, the domain of $h^{-1}(x)$ is the range of $h(x)$ i.e., $x\in[0,25]$ and the range of $h^{-1}(x)$ is the domain of $h(x)$ i.e., $y\geq0$.

The inverse function $h^{-1}(x)=\frac{\sqrt{25 - x}}{4}$ is a square - root function. And it has limits on the domain because the expression under the square root must be non - negative, so $25−x\geq0$ or $x\leq25$, and also in the physical context of the height of the ball, $x\geq0$.

Step2: Analyze the nature of the inverse function

The inverse function $h^{-1}(x)=\frac{\sqrt{25 - x}}{4}$ is a square - root function. Since the domain of $h^{-1}(x)$ is restricted by the non - negativity of the expression inside the square root and the physical context of the problem (height of the ball above the ground), it has limits on the domain.

Answer:

The inverse function will be a square - root function and will have limits on the domain. Function $h^{-1}$ will give $h^{-1}(x)=\frac{\sqrt{25 - x}}{4}$, where $x\in[0,25]$.