Precalculus Examples

$f (x) = \frac{x - 4}{x^{2} + 8 x + 16}$

Step 1

Set the denominator in $\frac{x - 4}{x^{2} + 8 x + 16}$ equal to $0$ to find where the expression is undefined.

$x^{2} + 8 x + 16 = 0$

Step 2

Solve for $x$ .

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Step 2.1

Factor using the perfect square rule.

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Step 2.1.1

Rewrite $16$ as $4^{2}$ .

$x^{2} + 8 x + 4^{2} = 0$

Step 2.1.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$8 x = 2 \cdot x \cdot 4$

Step 2.1.3

Rewrite the polynomial.

$x^{2} + 2 \cdot x \cdot 4 + 4^{2} = 0$

Step 2.1.4

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = x$ and $b = 4$ .

${(x + 4)}^{2} = 0$

Step 2.2

Set the $x + 4$ equal to $0$ .

$x + 4 = 0$

Step 2.3

Subtract $4$ from both sides of the equation.

$x = - 4$

Step 3

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

Interval Notation:

$(- \infty, - 4) \cup (- 4, \infty)$

Set-Builder Notation:

${x | x \neq - 4}$

Step 4