Find the Domain and Range f(x)=1/2x^(3/2)

$f (x) = \frac{1}{2} x^{\frac{3}{2}}$

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

$\frac{1}{2} \sqrt{x^{3}}$

Set the radicand in $\sqrt{x^{3}}$ greater than or equal to $0$ to find where the expression is defined.

$x^{3} \geq 0$

Solve for $x$ .

Take the cube root of both sides of the inequality to eliminate the exponent on the left side.

$\sqrt[3]{x^{3}} \geq \sqrt[3]{0}$

Simplify the equation.

Simplify the left side.

Pull terms out from under the radical.

$x \geq \sqrt[3]{0}$

Simplify the right side.

Simplify $\sqrt[3]{0}$ .

Rewrite $0$ as $0^{3}$ .

$x \geq \sqrt[3]{0^{3}}$

Pull terms out from under the radical.

$x \geq 0$

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

Interval Notation:

$[0, \infty)$

Set-Builder Notation:

${x | x \geq 0}$

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

Interval Notation:

$[0, \infty)$

Set-Builder Notation:

${y | y \geq 0}$

Determine the domain and range.

Domain: $[0, \infty), {x | x \geq 0}$

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