We will discuss the domain and range separately.

Domain

The domain $D$ is the set of all $x -$ values, or inputs, for which a function is defined. To identify the domain of a radical function, we should consider whether the radical's index is even or odd.

Even indices

For even indices, the domain is restricted to non-negative values. To find the domain of such a function, we set the radicand equal to zero and solve for

x .

Depending on the function, the solution will show the domain's upper or lower limit. Let's show a couple of examples.

f (x) = g (x) = - x + 4 : x + 4 = 0 \Leftrightarrow x = - 4 2 - x : 2 - x = 0 \Leftrightarrow x = 2

Let's graph the different functions.

Let's summarize the domain for

f (x)

and

g (x) .

f (x) : g (x) : x \geq - 4 x \leq 2

Odd indices

For odd indices the domain is not restricted in any way. All real numbers, positive, negative, and $0$ can be substituted into a radical function with an odd index. Let's show an example.

The domain for $h (x)$ is all real values of $x .$

Range

The range $R$ is the set of all $y -$ values, or outputs, a function gives.

Even indices

A radical function is either constantly increasing or decreasing. Therefore, to find the range of a radical function with an even index we should find the corresponding $y -$ value for the upper or lower limit of the domain, then graph the function. The shape it takes will tell us the range of the function. Below, we see a couple of examples.

Let's summarize the range for

f (x)

and

g (x) .

f (x) : g (x) : y \geq - 2 y \leq 2

Odd indices

Just like with the domain, the range of radical functions with odd indices are not restricted in either direction.

The range for $h (x)$ is all real values of $y .$

Exercise

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