Precalculus Examples

$f (x) = \frac{1}{x^{2} - 9}$ , $g (x) = \sqrt{x - 2}$

Step 1

Find the quotient of the functions.

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Step 1.1

Replace the function designators with the actual functions in $\frac{f (x)}{g (x)}$ .

$\frac{\frac{1}{x^{2} - 9}}{\sqrt{x - 2}}$

Step 1.2

Simplify.

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Step 1.2.1

Multiply the numerator by the reciprocal of the denominator.

$\frac{1}{x^{2} - 9} \cdot \frac{1}{\sqrt{x - 2}}$

Step 1.2.2

Simplify the denominator.

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Step 1.2.2.1

Rewrite $9$ as $3^{2}$ .

$\frac{1}{x^{2} - 3^{2}} \cdot \frac{1}{\sqrt{x - 2}}$

Step 1.2.2.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 3$ .

$\frac{1}{(x + 3) (x - 3)} \cdot \frac{1}{\sqrt{x - 2}}$

Step 1.2.3

Multiply $\frac{1}{\sqrt{x - 2}}$ by $\frac{\sqrt{x - 2}}{\sqrt{x - 2}}$ .

$\frac{1}{(x + 3) (x - 3)} (\frac{1}{\sqrt{x - 2}} \cdot \frac{\sqrt{x - 2}}{\sqrt{x - 2}})$

Step 1.2.4

Combine and simplify the denominator.

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Step 1.2.4.1

Multiply $\frac{1}{\sqrt{x - 2}}$ by $\frac{\sqrt{x - 2}}{\sqrt{x - 2}}$ .

$\frac{1}{(x + 3) (x - 3)} \cdot \frac{\sqrt{x - 2}}{\sqrt{x - 2} \sqrt{x - 2}}$

Step 1.2.4.2

Raise $\sqrt{x - 2}$ to the power of $1$ .

$\frac{1}{(x + 3) (x - 3)} \cdot \frac{\sqrt{x - 2}}{{\sqrt{x - 2}}^{1} \sqrt{x - 2}}$

Step 1.2.4.3

Raise $\sqrt{x - 2}$ to the power of $1$ .

$\frac{1}{(x + 3) (x - 3)} \cdot \frac{\sqrt{x - 2}}{{\sqrt{x - 2}}^{1} {\sqrt{x - 2}}^{1}}$

Step 1.2.4.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{1}{(x + 3) (x - 3)} \cdot \frac{\sqrt{x - 2}}{{\sqrt{x - 2}}^{1 + 1}}$

Step 1.2.4.5

Add $1$ and $1$ .

$\frac{1}{(x + 3) (x - 3)} \cdot \frac{\sqrt{x - 2}}{{\sqrt{x - 2}}^{2}}$

Step 1.2.4.6

Rewrite ${\sqrt{x - 2}}^{2}$ as $x - 2$ .

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Step 1.2.4.6.1

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

$\frac{1}{(x + 3) (x - 3)} \cdot \frac{\sqrt{x - 2}}{{({(x - 2)}^{\frac{1}{2}})}^{2}}$

Step 1.2.4.6.2

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

$\frac{1}{(x + 3) (x - 3)} \cdot \frac{\sqrt{x - 2}}{{(x - 2)}^{\frac{1}{2} \cdot 2}}$

Step 1.2.4.6.3

Combine $\frac{1}{2}$ and $2$ .

$\frac{1}{(x + 3) (x - 3)} \cdot \frac{\sqrt{x - 2}}{{(x - 2)}^{\frac{2}{2}}}$

Step 1.2.4.6.4

Cancel the common factor of $2$ .

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Step 1.2.4.6.4.1

Cancel the common factor.

$\frac{1}{(x + 3) (x - 3)} \cdot \frac{\sqrt{x - 2}}{{(x - 2)}^{\frac{2}{2}}}$

Step 1.2.4.6.4.2

Rewrite the expression.

$\frac{1}{(x + 3) (x - 3)} \cdot \frac{\sqrt{x - 2}}{{(x - 2)}^{1}}$

Step 1.2.4.6.5

Simplify.

$\frac{1}{(x + 3) (x - 3)} \cdot \frac{\sqrt{x - 2}}{x - 2}$

Step 1.2.5

Multiply $\frac{1}{(x + 3) (x - 3)}$ by $\frac{\sqrt{x - 2}}{x - 2}$ .

$\frac{\sqrt{x - 2}}{(x + 3) (x - 3) (x - 2)}$

Step 2

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

$x - 2 \geq 0$

Step 3

Add $2$ to both sides of the inequality.

$x \geq 2$

Step 4

Set the denominator in $\frac{\sqrt{x - 2}}{(x + 3) (x - 3) (x - 2)}$ equal to $0$ to find where the expression is undefined.

$(x + 3) (x - 3) (x - 2) = 0$

Step 5

Solve for $x$ .

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Step 5.1

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x + 3 = 0$

$x - 3 = 0$

$x - 2 = 0$

Step 5.2

Set $x + 3$ equal to $0$ and solve for $x$ .

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Step 5.2.1

Set $x + 3$ equal to $0$ .

$x + 3 = 0$

Step 5.2.2

Subtract $3$ from both sides of the equation.

$x = - 3$

Step 5.3

Set $x - 3$ equal to $0$ and solve for $x$ .

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Step 5.3.1

Set $x - 3$ equal to $0$ .

$x - 3 = 0$

Step 5.3.2

Add $3$ to both sides of the equation.

$x = 3$

Step 5.4

Set $x - 2$ equal to $0$ and solve for $x$ .

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Step 5.4.1

Set $x - 2$ equal to $0$ .

$x - 2 = 0$

Step 5.4.2

Add $2$ to both sides of the equation.

$x = 2$

Step 5.5

The final solution is all the values that make $(x + 3) (x - 3) (x - 2) = 0$ true.

$x = - 3, 3, 2$

Step 6

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

Interval Notation:

$(2, 3) \cup (3, \infty)$

Set-Builder Notation:

${x | x > 2, x \neq 3}$

Step 7