Precalculus Examples

$f (x) = \frac{2 x + 9}{x^{2} - 64}$ , $g (x) = \frac{x + 1}{x^{2} - 64}$

Step 1

Find the quotient of the functions.

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Step 1.1

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

$\frac{\frac{2 x + 9}{x^{2} - 64}}{\frac{x + 1}{x^{2} - 64}}$

Step 1.2

Simplify.

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Step 1.2.1

Multiply the numerator by the reciprocal of the denominator.

$\frac{2 x + 9}{x^{2} - 64} \cdot \frac{x^{2} - 64}{x + 1}$

Step 1.2.2

Cancel the common factor of $x^{2} - 64$ .

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Step 1.2.2.1

Cancel the common factor.

$\frac{2 x + 9}{x^{2} - 64} \cdot \frac{x^{2} - 64}{x + 1}$

Step 1.2.2.2

Rewrite the expression.

$(2 x + 9) \frac{1}{x + 1}$

Step 1.2.3

Multiply $2 x + 9$ by $\frac{1}{x + 1}$ .

$\frac{2 x + 9}{x + 1}$

Step 2

Set the denominator in $\frac{2 x + 9}{x + 1}$ equal to $0$ to find where the expression is undefined.

$x + 1 = 0$

Step 3

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 4

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

Interval Notation:

$(- \infty, - 1) \cup (- 1, \infty)$

Set-Builder Notation:

${x | x \neq - 1}$

Step 5