Precalculus Examples

$f (x) = x \sqrt{8 - x^{2}}$

Step 1

Find $f (- x)$ .

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Step 1.1

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

$f (- x) = (- x) \sqrt{8 - {(- x)}^{2}}$

Step 1.2

Apply the product rule to $- x$ .

$f (- x) = - x \sqrt{8 - ({(- 1)}^{2} x^{2})}$

Step 1.3

Multiply $- 1$ by ${(- 1)}^{2}$ by adding the exponents.

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Step 1.3.1

Move ${(- 1)}^{2}$ .

$f (- x) = - x \sqrt{8 + {(- 1)}^{2} \cdot (- 1 x^{2})}$

Step 1.3.2

Multiply ${(- 1)}^{2}$ by $- 1$ .

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Step 1.3.2.1

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

$f (- x) = - x \sqrt{8 + {(- 1)}^{2} \cdot ((- 1) x^{2})}$

Step 1.3.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f (- x) = - x \sqrt{8 + {(- 1)}^{2 + 1} x^{2}}$

Step 1.3.3

Add $2$ and $1$ .

$f (- x) = - x \sqrt{8 + {(- 1)}^{3} x^{2}}$

Step 1.4

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

$f (- x) = - x \sqrt{8 - x^{2}}$

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 $- x \sqrt{8 - x^{2}}$ $\neq$ $x \sqrt{8 - x^{2}}$ , 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

Remove parentheses.

$- f (x) = - x \sqrt{8 - x^{2}}$

Step 3.2

Since $- x \sqrt{8 - x^{2}} = - x \sqrt{8 - x^{2}}$ , the function is odd.

The function is odd

Step 4