Precalculus Examples

$f (x) = \frac{2}{x + 4}$ , $g (x) = x^{2} - 2$

Step 1

Replace the function designators with the actual functions in $f (x) + g (x)$ .

$\frac{2}{x + 4} + x^{2} - 2$

Step 2

Simplify.

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Step 2.1

To write $x^{2}$ as a fraction with a common denominator, multiply by $\frac{x + 4}{x + 4}$ .

$\frac{2}{x + 4} + \frac{x^{2} (x + 4)}{x + 4} - 2$

Step 2.2

Combine the numerators over the common denominator.

$\frac{2 + x^{2} (x + 4)}{x + 4} - 2$

Step 2.3

Simplify the numerator.

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Step 2.3.1

Apply the distributive property.

$\frac{2 + x^{2} x + x^{2} \cdot 4}{x + 4} - 2$

Step 2.3.2

Multiply $x^{2}$ by $x$ by adding the exponents.

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Step 2.3.2.1

Multiply $x^{2}$ by $x$ .

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Step 2.3.2.1.1

Raise $x$ to the power of $1$ .

$\frac{2 + x^{2} x^{1} + x^{2} \cdot 4}{x + 4} - 2$

Step 2.3.2.1.2

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

$\frac{2 + x^{2 + 1} + x^{2} \cdot 4}{x + 4} - 2$

Step 2.3.2.2

Add $2$ and $1$ .

$\frac{2 + x^{3} + x^{2} \cdot 4}{x + 4} - 2$

Step 2.3.3

Move $4$ to the left of $x^{2}$ .

$\frac{2 + x^{3} + 4 \cdot x^{2}}{x + 4} - 2$

Step 2.3.4

Reorder terms.

$\frac{x^{3} + 4 x^{2} + 2}{x + 4} - 2$

Step 2.4

To write $- 2$ as a fraction with a common denominator, multiply by $\frac{x + 4}{x + 4}$ .

$\frac{x^{3} + 4 x^{2} + 2}{x + 4} - 2 \cdot \frac{x + 4}{x + 4}$

Step 2.5

Simplify terms.

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Step 2.5.1

Combine $- 2$ and $\frac{x + 4}{x + 4}$ .

$\frac{x^{3} + 4 x^{2} + 2}{x + 4} + \frac{- 2 (x + 4)}{x + 4}$

Step 2.5.2

Combine the numerators over the common denominator.

$\frac{x^{3} + 4 x^{2} + 2 - 2 (x + 4)}{x + 4}$

Step 2.6

Simplify the numerator.

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Step 2.6.1

Apply the distributive property.

$\frac{x^{3} + 4 x^{2} + 2 - 2 x - 2 \cdot 4}{x + 4}$

Step 2.6.2

Multiply $- 2$ by $4$ .

$\frac{x^{3} + 4 x^{2} + 2 - 2 x - 8}{x + 4}$

Step 2.6.3

Subtract $8$ from $2$ .

$\frac{x^{3} + 4 x^{2} - 2 x - 6}{x + 4}$