Skill: Determine the domain and range of a function

The domain of a function is the set of allowed input values, or $x$-values. The range is the set of output values, or $y$-values, the function actually produces.

Consider the function

$$f(x)=\sqrt{16-x^2}.$$

Step 1: Find the domain

Because the expression is under an even root, the radicand must be nonnegative:

$$16-x^2\ge 0.$$

Rearrange:

$$x^2\le 16.$$

Therefore,

$$-4\le x\le 4.$$

So the domain is

$$\boxed{[-4,4]}.$$

Step 2: Find the range

The function is a square root, so its outputs cannot be negative:

$$f(x)\ge 0.$$

The greatest value occurs when $x^2$ is smallest. The smallest possible value of $x^2$ is $0$, which happens when $x=0$:

$$f(0)=\sqrt{16}=4.$$

Thus, the function’s outputs range from $0$ to $4$:

$$\boxed{[0,4]}.$$

The value $0$ occurs when $x=-4$ or $x=4$, and the value $4$ occurs when $x=0$.

Therefore, for

$$f(x)=\sqrt{16-x^2},$$

the domain is

$$\boxed{[-4,4]}$$

and the range is

$$\boxed{[0,4]}.$$