Precalculus Examples

$f (x) = \sqrt{x^{2} - 4}$ , $g (x) = \frac{x^{2}}{x^{2} + 1}$

Step 1

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

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

Step 2

Simplify.

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Step 2.1

Simplify each term.

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Step 2.1.1

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

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

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 $b = 2$ .

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

Step 2.2

To write $\sqrt{(x + 2) (x - 2)}$ as a fraction with a common denominator, multiply by $\frac{x^{2} + 1}{x^{2} + 1}$ .

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

Step 2.3

Combine the numerators over the common denominator.

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

Step 2.4

Simplify the numerator.

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Step 2.4.1

Apply the distributive property.

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

Step 2.4.2

Multiply $\sqrt{(x + 2) (x - 2)}$ by $1$ .

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