Precalculus Examples

$f (x) = - 6 x^{2} + x^{4} + 7 x^{6}$

Step 1

Find $f (- x)$ .

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Step 1.1

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

$f (- x) = - 6 {(- x)}^{2} + {(- x)}^{4} + 7 {(- x)}^{6}$

Step 1.2

Simplify each term.

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Step 1.2.1

Apply the product rule to $- x$ .

$f (- x) = - 6 ({(- 1)}^{2} x^{2}) + {(- x)}^{4} + 7 {(- x)}^{6}$

Step 1.2.2

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

$f (- x) = - 6 (1 x^{2}) + {(- x)}^{4} + 7 {(- x)}^{6}$

Step 1.2.3

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

$f (- x) = - 6 x^{2} + {(- x)}^{4} + 7 {(- x)}^{6}$

Step 1.2.4

Apply the product rule to $- x$ .

$f (- x) = - 6 x^{2} + {(- 1)}^{4} x^{4} + 7 {(- x)}^{6}$

Step 1.2.5

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

$f (- x) = - 6 x^{2} + 1 x^{4} + 7 {(- x)}^{6}$

Step 1.2.6

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

$f (- x) = - 6 x^{2} + x^{4} + 7 {(- x)}^{6}$

Step 1.2.7

Apply the product rule to $- x$ .

$f (- x) = - 6 x^{2} + x^{4} + 7 ({(- 1)}^{6} x^{6})$

Step 1.2.8

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

$f (- x) = - 6 x^{2} + x^{4} + 7 (1 x^{6})$

Step 1.2.9

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

$f (- x) = - 6 x^{2} + x^{4} + 7 x^{6}$

Step 2

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

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Step 2.1

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

Step 2.2

Since $- 6 x^{2} + x^{4} + 7 x^{6} = - 6 x^{2} + x^{4} + 7 x^{6}$ , the function is even.

The function is even

Step 3