Find the Domain and Range f(x)=(x-4)^(1/2)

$f (x) = {(x - 4)}^{\frac{1}{2}}$

Step 1

Convert expressions with fractional exponents to radicals.

Apply the rule $x^{\frac{m}{n}} = \sqrt[n]{x^{m}}$ to rewrite the exponentiation as a radical.

$\sqrt{{(x - 4)}^{1}}$

Anything raised to $1$ is the base itself.

$\sqrt{x - 4}$

Step 2

Set the radicand in $\sqrt{x - 4}$ greater than or equal to $0$ to find where the expression is defined.

$x - 4 \geq 0$

Step 3

Add $4$ to both sides of the inequality.

$x \geq 4$

Step 4

The domain is all values of $x$ that make the expression defined.

Interval Notation:

$[4, \infty)$

Set-Builder Notation:

${x | x \geq 4}$

Step 5

The range is the set of all valid $y$ values. Use the graph to find the range.

Interval Notation:

$[0, \infty)$

Set-Builder Notation:

${y | y \geq 0}$

Step 6

Determine the domain and range.

Domain: $[4, \infty), {x | x \geq 4}$

Range: $[0, \infty), {y | y \geq 0}$

Step 7