Algebra Examples

$\frac{1}{x + 5}$

Write $\frac{1}{x + 5}$ as a function.

$f (x) = \frac{1}{x + 5}$

Find $f (- x)$ .

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

$f (- x) = \frac{1}{(- x) + 5}$

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

$f (- x) = \frac{1}{- (x) + 5}$

Rewrite $5$ as $- 1 (- 5)$ .

$f (- x) = \frac{1}{- (x) - 1 \cdot - 5}$

Factor $- 1$ out of $- (x) - 1 (- 5)$ .

$f (- x) = \frac{1}{- (x - 5)}$

Rewrite negatives.

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Rewrite $- (x - 5)$ as $- 1 (x - 5)$ .

$f (- x) = \frac{1}{- 1 (x - 5)}$

Move the negative in front of the fraction.

$f (- x) = - \frac{1}{x - 5}$

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

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

$- \frac{1}{x - 5}$ $\neq$ $\frac{1}{x + 5}$

Since $- \frac{1}{x - 5}$ $\neq$ $\frac{1}{x + 5}$ , the function is not even.

The function is not even

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

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Multiply $- 1$ by $\frac{1}{x + 5}$ .

$- f (x) = - \frac{1}{x + 5}$

Since $- \frac{1}{x - 5}$ $\neq$ $- \frac{1}{x + 5}$ , the function is not odd.

The function is not odd

The function is neither odd nor even