Precalculus Examples

$g (x) = \frac{8 x}{\sqrt{2 x + 6}}$

Step 1

Set the radicand in $\sqrt{2 x + 6}$ greater than or equal to $0$ to find where the expression is defined.

$2 x + 6 \geq 0$

Step 2

Solve for $x$ .

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Step 2.1

Subtract $6$ from both sides of the inequality.

$2 x \geq - 6$

Step 2.2

Divide each term in $2 x \geq - 6$ by $2$ and simplify.

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Step 2.2.1

Divide each term in $2 x \geq - 6$ by $2$ .

$\frac{2 x}{2} \geq \frac{- 6}{2}$

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

Cancel the common factor of $2$ .

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Step 2.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} \geq \frac{- 6}{2}$

Step 2.2.2.1.2

Divide $x$ by $1$ .

$x \geq \frac{- 6}{2}$

Step 2.2.3

Simplify the right side.

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Step 2.2.3.1

Divide $- 6$ by $2$ .

$x \geq - 3$

Step 3

Set the denominator in $\frac{8 x}{\sqrt{2 x + 6}}$ equal to $0$ to find where the expression is undefined.

$\sqrt{2 x + 6} = 0$

Step 4

Solve for $x$ .

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Step 4.1

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

${\sqrt{2 x + 6}}^{2} = 0^{2}$

Step 4.2

Simplify each side of the equation.

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Step 4.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2 x + 6}$ as ${(2 x + 6)}^{\frac{1}{2}}$ .

${({(2 x + 6)}^{\frac{1}{2}})}^{2} = 0^{2}$

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Simplify ${({(2 x + 6)}^{\frac{1}{2}})}^{2}$ .

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Step 4.2.2.1.1

Multiply the exponents in ${({(2 x + 6)}^{\frac{1}{2}})}^{2}$ .

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Step 4.2.2.1.1.1

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

${(2 x + 6)}^{\frac{1}{2} \cdot 2} = 0^{2}$

Step 4.2.2.1.1.2

Cancel the common factor of $2$ .

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Step 4.2.2.1.1.2.1

Cancel the common factor.

${(2 x + 6)}^{\frac{1}{2} \cdot 2} = 0^{2}$

Step 4.2.2.1.1.2.2

Rewrite the expression.

${(2 x + 6)}^{1} = 0^{2}$

Step 4.2.2.1.2

Simplify.

$2 x + 6 = 0^{2}$

Step 4.2.3

Simplify the right side.

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Step 4.2.3.1

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

$2 x + 6 = 0$

Step 4.3

Solve for $x$ .

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Step 4.3.1

Subtract $6$ from both sides of the equation.

$2 x = - 6$

Step 4.3.2

Divide each term in $2 x = - 6$ by $2$ and simplify.

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Step 4.3.2.1

Divide each term in $2 x = - 6$ by $2$ .

$\frac{2 x}{2} = \frac{- 6}{2}$

Step 4.3.2.2

Simplify the left side.

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Step 4.3.2.2.1

Cancel the common factor of $2$ .

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Step 4.3.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{- 6}{2}$

Step 4.3.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 6}{2}$

Step 4.3.2.3

Simplify the right side.

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Step 4.3.2.3.1

Divide $- 6$ by $2$ .

$x = - 3$

Step 5

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

Interval Notation:

$(- 3, \infty)$

Set-Builder Notation:

${x | x > - 3}$

Step 6