Precalculus Examples

$f (x) = \frac{\log (x - 4)}{\sqrt{27 - 3 x}}$

Step 1

Set the argument in $\log (x - 4)$ greater than $0$ to find where the expression is defined.

$x - 4 > 0$

Step 2

Add $4$ to both sides of the inequality.

$x > 4$

Step 3

Set the radicand in $\sqrt{27 - 3 x}$ greater than or equal to $0$ to find where the expression is defined.

$27 - 3 x \geq 0$

Step 4

Solve for $x$ .

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Step 4.1

Subtract $27$ from both sides of the inequality.

$- 3 x \geq - 27$

Step 4.2

Divide each term in $- 3 x \geq - 27$ by $- 3$ and simplify.

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Step 4.2.1

Divide each term in $- 3 x \geq - 27$ by $- 3$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 3 x}{- 3} \leq \frac{- 27}{- 3}$

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Cancel the common factor of $- 3$ .

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Step 4.2.2.1.1

Cancel the common factor.

$\frac{- 3 x}{- 3} \leq \frac{- 27}{- 3}$

Step 4.2.2.1.2

Divide $x$ by $1$ .

$x \leq \frac{- 27}{- 3}$

Step 4.2.3

Simplify the right side.

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Step 4.2.3.1

Divide $- 27$ by $- 3$ .

$x \leq 9$

Step 5

Set the denominator in $\frac{\log (x - 4)}{\sqrt{27 - 3 x}}$ equal to $0$ to find where the expression is undefined.

$\sqrt{27 - 3 x} = 0$

Step 6

Solve for $x$ .

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Step 6.1

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

${\sqrt{27 - 3 x}}^{2} = 0^{2}$

Step 6.2

Simplify each side of the equation.

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Step 6.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{27 - 3 x}$ as ${(27 - 3 x)}^{\frac{1}{2}}$ .

${({(27 - 3 x)}^{\frac{1}{2}})}^{2} = 0^{2}$

Step 6.2.2

Simplify the left side.

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Step 6.2.2.1

Simplify ${({(27 - 3 x)}^{\frac{1}{2}})}^{2}$ .

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Step 6.2.2.1.1

Multiply the exponents in ${({(27 - 3 x)}^{\frac{1}{2}})}^{2}$ .

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Step 6.2.2.1.1.1

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

${(27 - 3 x)}^{\frac{1}{2} \cdot 2} = 0^{2}$

Step 6.2.2.1.1.2

Cancel the common factor of $2$ .

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Step 6.2.2.1.1.2.1

Cancel the common factor.

${(27 - 3 x)}^{\frac{1}{2} \cdot 2} = 0^{2}$

Step 6.2.2.1.1.2.2

Rewrite the expression.

${(27 - 3 x)}^{1} = 0^{2}$

Step 6.2.2.1.2

Simplify.

$27 - 3 x = 0^{2}$

Step 6.2.3

Simplify the right side.

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Step 6.2.3.1

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

$27 - 3 x = 0$

Step 6.3

Solve for $x$ .

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Step 6.3.1

Subtract $27$ from both sides of the equation.

$- 3 x = - 27$

Step 6.3.2

Divide each term in $- 3 x = - 27$ by $- 3$ and simplify.

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Step 6.3.2.1

Divide each term in $- 3 x = - 27$ by $- 3$ .

$\frac{- 3 x}{- 3} = \frac{- 27}{- 3}$

Step 6.3.2.2

Simplify the left side.

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Step 6.3.2.2.1

Cancel the common factor of $- 3$ .

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Step 6.3.2.2.1.1

Cancel the common factor.

$\frac{- 3 x}{- 3} = \frac{- 27}{- 3}$

Step 6.3.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 27}{- 3}$

Step 6.3.2.3

Simplify the right side.

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Step 6.3.2.3.1

Divide $- 27$ by $- 3$ .

$x = 9$

Step 7

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

Interval Notation:

$(4, 9)$

Set-Builder Notation:

${x | 4 < x < 9}$

Step 8