Precalculus Examples

$f (x) = 2 x \sqrt{x^{2} + 3}$

Step 1

Find $f (- x)$ .

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Step 1.1

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

$f (- x) = 2 (- x) \sqrt{{(- x)}^{2} + 3}$

Step 1.2

Multiply $- 1$ by $2$ .

$f (- x) = - 2 x \sqrt{{(- x)}^{2} + 3}$

Step 1.3

Apply the product rule to $- x$ .

$f (- x) = - 2 x \sqrt{{(- 1)}^{2} x^{2} + 3}$

Step 1.4

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

$f (- x) = - 2 x \sqrt{1 x^{2} + 3}$

Step 1.5

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

$f (- x) = - 2 x \sqrt{x^{2} + 3}$

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 $- 2 x \sqrt{x^{2} + 3}$ $\neq$ $2 x \sqrt{x^{2} + 3}$ , the function is not even.

The function is not even

Step 3

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

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Step 3.1

Multiply $2$ by $- 1$ .

$- f (x) = - 2 x \sqrt{x^{2} + 3}$

Step 3.2

Since $- 2 x \sqrt{x^{2} + 3} = - 2 x \sqrt{x^{2} + 3}$ , the function is odd.

The function is odd

Step 4