Determine the domain and range of a function

A function’s domain is the set of allowable input values, and its range is the set of output values the function actually produces; these sets can be identified from equations, tables, graphs, or contextual constraints. Determining them requires recognizing restrictions such as zero denominators and negative radicands for even roots, and analyzing how outputs vary rather than assuming domain and range are both all real numbers. This understanding supports function composition, inverses, and mathematical modeling; complex-valued and more abstract domain–codomain treatments are not included.

Detailed Explanation: 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]}.

Learn by doing: Determine the domain and range of a function

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Function Domain - Fraction Root of Linear over Linear to Number Line

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