Precalculus Examples

$f (x) = \frac{4 x^{2} + 1}{x^{2} + x + 16}$

Step 1

Set the denominator in $\frac{4 x^{2} + 1}{x^{2} + x + 16}$ equal to $0$ to find where the expression is undefined.

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

Step 2

Solve for $x$ .

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Step 2.1

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 2.2

Substitute the values $a = 1$ , $b = 1$ , and $c = 16$ into the quadratic formula and solve for $x$ .

$\frac{- 1 \pm \sqrt{1^{2} - 4 \cdot (1 \cdot 16)}}{2 \cdot 1}$

Step 2.3

Simplify.

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Step 2.3.1

Simplify the numerator.

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Step 2.3.1.1

One to any power is one.

$x = \frac{- 1 \pm \sqrt{1 - 4 \cdot 1 \cdot 16}}{2 \cdot 1}$

Step 2.3.1.2

Multiply $- 4 \cdot 1 \cdot 16$ .

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Step 2.3.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 1 \pm \sqrt{1 - 4 \cdot 16}}{2 \cdot 1}$

Step 2.3.1.2.2

Multiply $- 4$ by $16$ .

$x = \frac{- 1 \pm \sqrt{1 - 64}}{2 \cdot 1}$

Step 2.3.1.3

Subtract $64$ from $1$ .

$x = \frac{- 1 \pm \sqrt{- 63}}{2 \cdot 1}$

Step 2.3.1.4

Rewrite $- 63$ as $- 1 (63)$ .

$x = \frac{- 1 \pm \sqrt{- 1 \cdot 63}}{2 \cdot 1}$

Step 2.3.1.5

Rewrite $\sqrt{- 1 (63)}$ as $\sqrt{- 1} \cdot \sqrt{63}$ .

$x = \frac{- 1 \pm \sqrt{- 1} \cdot \sqrt{63}}{2 \cdot 1}$

Step 2.3.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{- 1 \pm i \cdot \sqrt{63}}{2 \cdot 1}$

Step 2.3.1.7

Rewrite $63$ as $3^{2} \cdot 7$ .

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Step 2.3.1.7.1

Factor $9$ out of $63$ .

$x = \frac{- 1 \pm i \cdot \sqrt{9 (7)}}{2 \cdot 1}$

Step 2.3.1.7.2

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

$x = \frac{- 1 \pm i \cdot \sqrt{3^{2} \cdot 7}}{2 \cdot 1}$

Step 2.3.1.8

Pull terms out from under the radical.

$x = \frac{- 1 \pm i \cdot (3 \sqrt{7})}{2 \cdot 1}$

Step 2.3.1.9

Move $3$ to the left of $i$ .

$x = \frac{- 1 \pm 3 i \sqrt{7}}{2 \cdot 1}$

Step 2.3.2

Multiply $2$ by $1$ .

$x = \frac{- 1 \pm 3 i \sqrt{7}}{2}$

Step 2.4

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 2.4.1

Simplify the numerator.

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Step 2.4.1.1

One to any power is one.

$x = \frac{- 1 \pm \sqrt{1 - 4 \cdot 1 \cdot 16}}{2 \cdot 1}$

Step 2.4.1.2

Multiply $- 4 \cdot 1 \cdot 16$ .

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Step 2.4.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 1 \pm \sqrt{1 - 4 \cdot 16}}{2 \cdot 1}$

Step 2.4.1.2.2

Multiply $- 4$ by $16$ .

$x = \frac{- 1 \pm \sqrt{1 - 64}}{2 \cdot 1}$

Step 2.4.1.3

Subtract $64$ from $1$ .

$x = \frac{- 1 \pm \sqrt{- 63}}{2 \cdot 1}$

Step 2.4.1.4

Rewrite $- 63$ as $- 1 (63)$ .

$x = \frac{- 1 \pm \sqrt{- 1 \cdot 63}}{2 \cdot 1}$

Step 2.4.1.5

Rewrite $\sqrt{- 1 (63)}$ as $\sqrt{- 1} \cdot \sqrt{63}$ .

$x = \frac{- 1 \pm \sqrt{- 1} \cdot \sqrt{63}}{2 \cdot 1}$

Step 2.4.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{- 1 \pm i \cdot \sqrt{63}}{2 \cdot 1}$

Step 2.4.1.7

Rewrite $63$ as $3^{2} \cdot 7$ .

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Step 2.4.1.7.1

Factor $9$ out of $63$ .

$x = \frac{- 1 \pm i \cdot \sqrt{9 (7)}}{2 \cdot 1}$

Step 2.4.1.7.2

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

$x = \frac{- 1 \pm i \cdot \sqrt{3^{2} \cdot 7}}{2 \cdot 1}$

Step 2.4.1.8

Pull terms out from under the radical.

$x = \frac{- 1 \pm i \cdot (3 \sqrt{7})}{2 \cdot 1}$

Step 2.4.1.9

Move $3$ to the left of $i$ .

$x = \frac{- 1 \pm 3 i \sqrt{7}}{2 \cdot 1}$

Step 2.4.2

Multiply $2$ by $1$ .

$x = \frac{- 1 \pm 3 i \sqrt{7}}{2}$

Step 2.4.3

Change the $\pm$ to $+$ .

$x = \frac{- 1 + 3 i \sqrt{7}}{2}$

Step 2.4.4

Rewrite $- 1$ as $- 1 (1)$ .

$x = \frac{- 1 \cdot 1 + 3 i \sqrt{7}}{2}$

Step 2.4.5

Factor $- 1$ out of $3 i \sqrt{7}$ .

$x = \frac{- 1 \cdot 1 - (- 3 i \sqrt{7})}{2}$

Step 2.4.6

Factor $- 1$ out of $- 1 (1) - (- 3 i \sqrt{7})$ .

$x = \frac{- 1 (1 - 3 i \sqrt{7})}{2}$

Step 2.4.7

Move the negative in front of the fraction.

$x = - \frac{1 - 3 i \sqrt{7}}{2}$

Step 2.5

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 2.5.1

Simplify the numerator.

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Step 2.5.1.1

One to any power is one.

$x = \frac{- 1 \pm \sqrt{1 - 4 \cdot 1 \cdot 16}}{2 \cdot 1}$

Step 2.5.1.2

Multiply $- 4 \cdot 1 \cdot 16$ .

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Step 2.5.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 1 \pm \sqrt{1 - 4 \cdot 16}}{2 \cdot 1}$

Step 2.5.1.2.2

Multiply $- 4$ by $16$ .

$x = \frac{- 1 \pm \sqrt{1 - 64}}{2 \cdot 1}$

Step 2.5.1.3

Subtract $64$ from $1$ .

$x = \frac{- 1 \pm \sqrt{- 63}}{2 \cdot 1}$

Step 2.5.1.4

Rewrite $- 63$ as $- 1 (63)$ .

$x = \frac{- 1 \pm \sqrt{- 1 \cdot 63}}{2 \cdot 1}$

Step 2.5.1.5

Rewrite $\sqrt{- 1 (63)}$ as $\sqrt{- 1} \cdot \sqrt{63}$ .

$x = \frac{- 1 \pm \sqrt{- 1} \cdot \sqrt{63}}{2 \cdot 1}$

Step 2.5.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{- 1 \pm i \cdot \sqrt{63}}{2 \cdot 1}$

Step 2.5.1.7

Rewrite $63$ as $3^{2} \cdot 7$ .

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Step 2.5.1.7.1

Factor $9$ out of $63$ .

$x = \frac{- 1 \pm i \cdot \sqrt{9 (7)}}{2 \cdot 1}$

Step 2.5.1.7.2

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

$x = \frac{- 1 \pm i \cdot \sqrt{3^{2} \cdot 7}}{2 \cdot 1}$

Step 2.5.1.8

Pull terms out from under the radical.

$x = \frac{- 1 \pm i \cdot (3 \sqrt{7})}{2 \cdot 1}$

Step 2.5.1.9

Move $3$ to the left of $i$ .

$x = \frac{- 1 \pm 3 i \sqrt{7}}{2 \cdot 1}$

Step 2.5.2

Multiply $2$ by $1$ .

$x = \frac{- 1 \pm 3 i \sqrt{7}}{2}$

Step 2.5.3

Change the $\pm$ to $-$ .

$x = \frac{- 1 - 3 i \sqrt{7}}{2}$

Step 2.5.4

Rewrite $- 1$ as $- 1 (1)$ .

$x = \frac{- 1 \cdot 1 - 3 i \sqrt{7}}{2}$

Step 2.5.5

Factor $- 1$ out of $- 3 i \sqrt{7}$ .

$x = \frac{- 1 \cdot 1 - (3 i \sqrt{7})}{2}$

Step 2.5.6

Factor $- 1$ out of $- 1 (1) - (3 i \sqrt{7})$ .

$x = \frac{- 1 (1 + 3 i \sqrt{7})}{2}$

Step 2.5.7

Move the negative in front of the fraction.

$x = - \frac{1 + 3 i \sqrt{7}}{2}$

Step 2.6

The final answer is the combination of both solutions.

$x = - \frac{1 - 3 i \sqrt{7}}{2}, - \frac{1 + 3 i \sqrt{7}}{2}$

Step 3

The domain is all real numbers.

Interval Notation:

$(- \infty, \infty)$

Set-Builder Notation:

${x | x \in ℝ}$

Step 4

The range is the set of all valid $y$ values. Use the graph to find the range.

Interval Notation:

$[\frac{130 - 2 \sqrt{3973}}{63}, \frac{130 + 2 \sqrt{3973}}{63}]$

Set-Builder Notation:

${y | \frac{130 - 2 \sqrt{3973}}{63} \leq y \leq \frac{130 + 2 \sqrt{3973}}{63}}$

Step 5

Determine the domain and range.

Domain: $(- \infty, \infty), {x | x \in ℝ}$

Range: $[\frac{130 - 2 \sqrt{3973}}{63}, \frac{130 + 2 \sqrt{3973}}{63}], {y | \frac{130 - 2 \sqrt{3973}}{63} \leq y \leq \frac{130 + 2 \sqrt{3973}}{63}}$

Step 6