Find the Domain and Range 5/( cube root of x-8)

$\frac{5}{\sqrt[3]{x - 8}}$

Step 1

Set the denominator in $\frac{5}{\sqrt[3]{x - 8}}$ equal to $0$ to find where the expression is undefined.

$\sqrt[3]{x - 8} = 0$

Step 2

Solve for $x$ .

To remove the radical on the left side of the equation, cube both sides of the equation.

${\sqrt[3]{x - 8}}^{3} = 0^{3}$

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{x - 8}$ as ${(x - 8)}^{\frac{1}{3}}$ .

${({(x - 8)}^{\frac{1}{3}})}^{3} = 0^{3}$

Simplify the left side.

Simplify ${({(x - 8)}^{\frac{1}{3}})}^{3}$ .

Multiply the exponents in ${({(x - 8)}^{\frac{1}{3}})}^{3}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

${(x - 8)}^{\frac{1}{3} \cdot 3} = 0^{3}$

Cancel the common factor of $3$ .

Cancel the common factor.

${(x - 8)}^{\frac{1}{3} \cdot 3} = 0^{3}$

Rewrite the expression.

${(x - 8)}^{1} = 0^{3}$

Simplify.

$x - 8 = 0^{3}$

Simplify the right side.

Raising $0$ to any positive power yields $0$ .

$x - 8 = 0$

Add $8$ to both sides of the equation.

$x = 8$

Step 3

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

Interval Notation:

$(- \infty, 8) \cup (8, \infty)$

Set-Builder Notation:

${x | x \neq 8}$

Step 4

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

Interval Notation:

$(- \infty, 0) \cup (0, \infty)$

Set-Builder Notation:

${y | y \neq 0}$

Step 5

Determine the domain and range.

Domain: $(- \infty, 8) \cup (8, \infty), {x | x \neq 8}$

Range: $(- \infty, 0) \cup (0, \infty), {y | y \neq 0}$

Step 6