Precalculus Examples

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

Step 1

Simplify each term.

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Step 1.1

Factor $x^{2}$ out of $x^{4} - x^{2}$ .

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Step 1.1.1

Factor $x^{2}$ out of $x^{4}$ .

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

Step 1.1.2

Factor $x^{2}$ out of $- x^{2}$ .

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

Step 1.1.3

Factor $x^{2}$ out of $x^{2} x^{2} + x^{2} \cdot - 1$ .

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

Step 1.2

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

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

Step 1.3

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 $b = 1$ .

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

Step 1.4

Rewrite $x^{2} (x + 1) (x - 1)$ as $x^{2} ((x + 1^{2}) (x - 1))$ .

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Step 1.4.1

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

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

Step 1.4.2

Add parentheses.

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

Step 1.5

Pull terms out from under the radical.

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

Step 1.6

One to any power is one.

$f (x) = x \sqrt{(x + 1) (x - 1)} + 4$

Step 2

Find $f (- x)$ .

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Step 2.1

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

$f (- x) = (- x) \sqrt{((- x) + 1) ((- x) - 1)} + 4$

Step 2.2

Remove parentheses.

$f (- x) = - x \sqrt{(- x + 1) (- x - 1)} + 4$

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 $- x \sqrt{(- x + 1) (- x - 1)} + 4 = x \sqrt{(x + 1) (x - 1)} + 4$ , the function is even.

The function is even

Step 4