Find the Domain and Range y=-2(3x)^(1/6)

$y = - 2 {(3 x)}^{\frac{1}{6}}$

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.

$- 2 \sqrt[6]{{(3 x)}^{1}}$

Anything raised to $1$ is the base itself.

$- 2 \sqrt[6]{3 x}$

Step 2

Set the radicand in $\sqrt[6]{3 x}$ greater than or equal to $0$ to find where the expression is defined.

$3 x \geq 0$

Step 3

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 $3$ .

$x \geq 0$

Step 4

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

Interval Notation:

$[0, \infty)$

Set-Builder Notation:

${x | x \geq 0}$

Step 5

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

Interval Notation:

$(- \infty, 0]$

Set-Builder Notation:

${y | y \leq 0}$

Step 6

Determine the domain and range.

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

Range: $(- \infty, 0], {y | y \leq 0}$

Step 7