Precalculus Examples

$f (x) = \frac{1}{5} x^{6} - 3 x^{2}$

Step 1

Combine $\frac{1}{5}$ and $x^{6}$ .

$f (x) = \frac{x^{6}}{5} - 3 x^{2}$

Step 2

Find $f (- x)$ .

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Step 2.1

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

$f (- x) = \frac{{(- x)}^{6}}{5} - 3 {(- x)}^{2}$

Step 2.2

Simplify each term.

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Step 2.2.1

Simplify the numerator.

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Step 2.2.1.1

Apply the product rule to $- x$ .

$f (- x) = \frac{{(- 1)}^{6} x^{6}}{5} - 3 {(- x)}^{2}$

Step 2.2.1.2

Raise $- 1$ to the power of $6$ .

$f (- x) = \frac{1 x^{6}}{5} - 3 {(- x)}^{2}$

Step 2.2.1.3

Multiply $x^{6}$ by $1$ .

$f (- x) = \frac{x^{6}}{5} - 3 {(- x)}^{2}$

Step 2.2.2

Apply the product rule to $- x$ .

$f (- x) = \frac{x^{6}}{5} - 3 ({(- 1)}^{2} x^{2})$

Step 2.2.3

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

$f (- x) = \frac{x^{6}}{5} - 3 (1 x^{2})$

Step 2.2.4

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

$f (- x) = \frac{x^{6}}{5} - 3 x^{2}$

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 $\frac{x^{6}}{5} - 3 x^{2} = \frac{x^{6}}{5} - 3 x^{2}$ , the function is even.

The function is even

Step 4