Precalculus Examples

$f (x) = \frac{1}{x^{2} - 1}$ , $g (x) = \sqrt{x + 3}$

Step 1

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

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

Step 2

Simplify.

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Step 2.1

Simplify the denominator.

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Step 2.1.1

Rewrite $1$ as $1^{2}$ .

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

Step 2.1.2

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 $b = 1$ .

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

Step 2.2

To write $\sqrt{x + 3}$ as a fraction with a common denominator, multiply by $\frac{(x + 1) (x - 1)}{(x + 1) (x - 1)}$ .

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

Step 2.3

Simplify terms.

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Step 2.3.1

Combine $\sqrt{x + 3}$ and $\frac{(x + 1) (x - 1)}{(x + 1) (x - 1)}$ .

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

Step 2.3.2

Combine the numerators over the common denominator.

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

Step 2.4

Simplify the numerator.

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Step 2.4.1

Expand $(x + 1) (x - 1)$ using the FOIL Method.

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Step 2.4.1.1

Apply the distributive property.

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

Step 2.4.1.2

Apply the distributive property.

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

Step 2.4.1.3

Apply the distributive property.

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

Step 2.4.2

Simplify and combine like terms.

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Step 2.4.2.1

Simplify each term.

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Step 2.4.2.1.1

Multiply $x$ by $x$ .

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

Step 2.4.2.1.2

Move $- 1$ to the left of $x$ .

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

Step 2.4.2.1.3

Rewrite $- 1 x$ as $- x$ .

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

Step 2.4.2.1.4

Multiply $x$ by $1$ .

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

Step 2.4.2.1.5

Multiply $- 1$ by $1$ .

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

Step 2.4.2.2

Add $- x$ and $x$ .

$\frac{1 + \sqrt{x + 3} (x^{2} + 0 - 1)}{(x + 1) (x - 1)}$

Step 2.4.2.3

Add $x^{2}$ and $0$ .

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

Step 2.4.3

Apply the distributive property.

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

Step 2.4.4

Move $- 1$ to the left of $\sqrt{x + 3}$ .

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

Step 2.4.5

Simplify each term.

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