Precalculus Examples

$g (x) = x^{2} \cos (x) + x^{3} \sin (x)$

Step 1

Find $f (- x)$ .

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Step 1.1

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

$g (- x) = {(- x)}^{2} \cos (- x) + {(- x)}^{3} \sin (- x)$

Step 1.2

Simplify each term.

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Step 1.2.1

Apply the product rule to $- x$ .

$g (- x) = {(- 1)}^{2} x^{2} \cos (- x) + {(- x)}^{3} \sin (- x)$

Step 1.2.2

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

$g (- x) = 1 x^{2} \cos (- x) + {(- x)}^{3} \sin (- x)$

Step 1.2.3

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

$g (- x) = x^{2} \cos (- x) + {(- x)}^{3} \sin (- x)$

Step 1.2.4

Since $\cos (- x)$ is an even function, rewrite $\cos (- x)$ as $\cos (x)$ .

$g (- x) = x^{2} \cos (x) + {(- x)}^{3} \sin (- x)$

Step 1.2.5

Apply the product rule to $- x$ .

$g (- x) = x^{2} \cos (x) + {(- 1)}^{3} x^{3} \sin (- x)$

Step 1.2.6

Raise $- 1$ to the power of $3$ .

$g (- x) = x^{2} \cos (x) - x^{3} \sin (- x)$

Step 1.2.7

Since $\sin (- x)$ is an odd function, rewrite $\sin (- x)$ as $- \sin (x)$ .

$g (- x) = x^{2} \cos (x) - x^{3} (- \sin (x))$

Step 1.2.8

Multiply $- x^{3} (- \sin (x))$ .

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Step 1.2.8.1

Multiply $- 1$ by $- 1$ .

$g (- x) = x^{2} \cos (x) + 1 x^{3} \sin (x)$

Step 1.2.8.2

Multiply $x^{3}$ by $1$ .

$g (- x) = x^{2} \cos (x) + x^{3} \sin (x)$

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 $x^{2} \cos (x) + x^{3} \sin (x) = x^{2} \cos (x) + x^{3} \sin (x)$ , the function is even.

The function is even

Step 3