Precalculus Examples

$f (x) = \frac{2 x + 4}{\sqrt{24 - 8 x}}$

Step 1

Set the radicand in $\sqrt{24 - 8 x}$ greater than or equal to $0$ to find where the expression is defined.

$24 - 8 x \geq 0$

Step 2

Solve for $x$ .

Tap for more steps...

Step 2.1

Subtract $24$ from both sides of the inequality.

$- 8 x \geq - 24$

Step 2.2

Divide each term in $- 8 x \geq - 24$ by $- 8$ and simplify.

Tap for more steps...

Step 2.2.1

Divide each term in $- 8 x \geq - 24$ by $- 8$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 8 x}{- 8} \leq \frac{- 24}{- 8}$

Step 2.2.2

Simplify the left side.

Tap for more steps...

Step 2.2.2.1

Cancel the common factor of $- 8$ .

Tap for more steps...

Step 2.2.2.1.1

Cancel the common factor.

$\frac{- 8 x}{- 8} \leq \frac{- 24}{- 8}$

Step 2.2.2.1.2

Divide $x$ by $1$ .

$x \leq \frac{- 24}{- 8}$

Step 2.2.3

Simplify the right side.

Tap for more steps...

Step 2.2.3.1

Divide $- 24$ by $- 8$ .

$x \leq 3$

Step 3

Set the denominator in $\frac{2 x + 4}{\sqrt{24 - 8 x}}$ equal to $0$ to find where the expression is undefined.

$\sqrt{24 - 8 x} = 0$

Step 4

Solve for $x$ .

Tap for more steps...

Step 4.1

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

${\sqrt{24 - 8 x}}^{2} = 0^{2}$

Step 4.2

Simplify each side of the equation.

Tap for more steps...

Step 4.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{24 - 8 x}$ as ${(24 - 8 x)}^{\frac{1}{2}}$ .

${({(24 - 8 x)}^{\frac{1}{2}})}^{2} = 0^{2}$

Step 4.2.2

Simplify the left side.

Tap for more steps...

Step 4.2.2.1

Simplify ${({(24 - 8 x)}^{\frac{1}{2}})}^{2}$ .

Tap for more steps...

Step 4.2.2.1.1

Multiply the exponents in ${({(24 - 8 x)}^{\frac{1}{2}})}^{2}$ .

Tap for more steps...

Step 4.2.2.1.1.1

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

${(24 - 8 x)}^{\frac{1}{2} \cdot 2} = 0^{2}$

Step 4.2.2.1.1.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 4.2.2.1.1.2.1

Cancel the common factor.

${(24 - 8 x)}^{\frac{1}{2} \cdot 2} = 0^{2}$

Step 4.2.2.1.1.2.2

Rewrite the expression.

${(24 - 8 x)}^{1} = 0^{2}$

Step 4.2.2.1.2

Simplify.

$24 - 8 x = 0^{2}$

Step 4.2.3

Simplify the right side.

Tap for more steps...

Step 4.2.3.1

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

$24 - 8 x = 0$

Step 4.3

Solve for $x$ .

Tap for more steps...

Step 4.3.1

Subtract $24$ from both sides of the equation.

$- 8 x = - 24$

Step 4.3.2

Divide each term in $- 8 x = - 24$ by $- 8$ and simplify.

Tap for more steps...

Step 4.3.2.1

Divide each term in $- 8 x = - 24$ by $- 8$ .

$\frac{- 8 x}{- 8} = \frac{- 24}{- 8}$

Step 4.3.2.2

Simplify the left side.

Tap for more steps...

Step 4.3.2.2.1

Cancel the common factor of $- 8$ .

Tap for more steps...

Step 4.3.2.2.1.1

Cancel the common factor.

$\frac{- 8 x}{- 8} = \frac{- 24}{- 8}$

Step 4.3.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 24}{- 8}$

Step 4.3.2.3

Simplify the right side.

Tap for more steps...

Step 4.3.2.3.1

Divide $- 24$ by $- 8$ .

$x = 3$

Step 5

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

Interval Notation:

$(- \infty, 3)$

Set-Builder Notation:

${x | x < 3}$

Step 6