Precalculus Examples

$f (x) = \sqrt{4 - x^{2}}$

Step 1

Simplify.

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Step 1.1

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

$f (x) = \sqrt{2^{2} - x^{2}}$

Step 1.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 2$ and $b = x$ .

$f (x) = \sqrt{(2 + x) (2 - x)}$

Step 2

Find $f (- x)$ .

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Step 2.1

Find $f (- x)$ by substituting $- x$ for all occurrence of $x$ in $f (x)$ .

$f (- x) = \sqrt{(2 - x) (2 - (- x))}$

Step 2.2

Remove parentheses.

$f (- x) = \sqrt{(2 - x) (2 - (- x))}$

Step 2.3

Multiply $- - x$ .

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Step 2.3.1

Multiply $- 1$ by $- 1$ .

$f (- x) = \sqrt{(2 - x) (2 + 1 x)}$

Step 2.3.2

Multiply $x$ by $1$ .

$f (- x) = \sqrt{(2 - x) (2 + x)}$

Step 3

A function is even if $f (- x) = f (x)$ .

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Step 3.1

Check if $f (- x) = f (x)$ .

Step 3.2

Since $\sqrt{(2 - x) (2 + x)} = \sqrt{(2 + x) (2 - x)}$ , the function is even.

The function is even

Step 4