Find the Domain and Range w(x)=-(3x)^(1/2)-4

$w (x) = - {(3 x)}^{\frac{1}{2}} - 4$

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.

$- \sqrt{{(3 x)}^{1}} - 4$

Anything raised to $1$ is the base itself.

$- \sqrt{3 x} - 4$

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

$3 x \geq 0$

Divide each term in $3 x \geq 0$ by $3$ and simplify.

Divide each term in $3 x \geq 0$ by $3$ .

$\frac{3 x}{3} \geq \frac{0}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 x}{3} \geq \frac{0}{3}$

Divide $x$ by $1$ .

$x \geq \frac{0}{3}$

Simplify the right side.

Divide $0$ by $3$ .

$x \geq 0$

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:

$(- \infty, - 4]$

Set-Builder Notation:

${y | y \leq - 4}$

Determine the domain and range.

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

Range: $(- \infty, - 4], {y | y \leq - 4}$