Precalculus Examples

$g (x) = \frac{1}{x + 4}$

Step 1

Find $f (- x)$ .

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Step 1.1

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

$g (- x) = \frac{1}{(- x) + 4}$

Step 1.2

Factor $- 1$ out of $- x$ .

$g (- x) = \frac{1}{- (x) + 4}$

Step 1.3

Rewrite $4$ as $- 1 (- 4)$ .

$g (- x) = \frac{1}{- (x) - 1 \cdot - 4}$

Step 1.4

Factor $- 1$ out of $- (x) - 1 (- 4)$ .

$g (- x) = \frac{1}{- (x - 4)}$

Step 1.5

Rewrite negatives.

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Step 1.5.1

Rewrite $- (x - 4)$ as $- 1 (x - 4)$ .

$g (- x) = \frac{1}{- 1 (x - 4)}$

Step 1.5.2

Move the negative in front of the fraction.

$g (- x) = - \frac{1}{x - 4}$

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 $- \frac{1}{x - 4}$ $\neq$ $\frac{1}{x + 4}$ , 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 $- 1$ by $\frac{1}{x + 4}$ .

$- f (x) = - \frac{1}{x + 4}$

Step 3.2

Since $- \frac{1}{x - 4}$ $\neq$ $- \frac{1}{x + 4}$ , the function is not odd.

The function is not odd

Step 4

The function is neither odd nor even

Step 5