Precalculus Examples

$f (x) = \frac{1}{x^{6} + 9 x^{4}}$

Step 1

Factor $x^{4}$ out of $x^{6} + 9 x^{4}$ .

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Step 1.1

Factor $x^{4}$ out of $x^{6}$ .

$f (x) = \frac{1}{x^{4} x^{2} + 9 x^{4}}$

Step 1.2

Factor $x^{4}$ out of $9 x^{4}$ .

$f (x) = \frac{1}{x^{4} x^{2} + x^{4} \cdot 9}$

Step 1.3

Factor $x^{4}$ out of $x^{4} x^{2} + x^{4} \cdot 9$ .

$f (x) = \frac{1}{x^{4} (x^{2} + 9)}$

Step 2

Find $f (- x)$ .

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Step 2.1

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

$f (- x) = \frac{1}{{(- x)}^{4} ({(- x)}^{2} + 9)}$

Step 2.2

Simplify the denominator.

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Step 2.2.1

Apply the product rule to $- x$ .

$f (- x) = \frac{1}{{(- 1)}^{4} x^{4} ({(- x)}^{2} + 9)}$

Step 2.2.2

Apply the product rule to $- x$ .

$f (- x) = \frac{1}{{(- 1)}^{4} x^{4} ({(- 1)}^{2} x^{2} + 9)}$

Step 2.2.3

Raise $- 1$ to the power of $2$ .

$f (- x) = \frac{1}{{(- 1)}^{4} x^{4} (1 x^{2} + 9)}$

Step 2.2.4

Multiply $x^{2}$ by $1$ .

$f (- x) = \frac{1}{{(- 1)}^{4} x^{4} (x^{2} + 9)}$

Step 2.2.5

Raise $- 1$ to the power of $4$ .

$f (- x) = \frac{1}{1 x^{4} (x^{2} + 9)}$

Step 2.2.6

Multiply $x^{4}$ by $1$ .

$f (- x) = \frac{1}{x^{4} (x^{2} + 9)}$

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 $\frac{1}{x^{4} (x^{2} + 9)} = \frac{1}{x^{4} (x^{2} + 9)}$ , the function is even.

The function is even

Step 4