Precalculus Examples

$f (x) = \frac{x}{x^{2} + 1}$

Step 1

Find $f (- x)$ .

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Step 1.1

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

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

Step 1.2

Simplify the denominator.

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Step 1.2.1

Apply the product rule to $- x$ .

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

Step 1.2.2

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

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

Step 1.2.3

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

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

Step 1.3

Move the negative in front of the fraction.

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

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 $- \frac{x}{x^{2} + 1}$ $\neq$ $\frac{x}{x^{2} + 1}$ , 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 $\frac{x}{x^{2} + 1}$ .

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

Step 3.2

Since $- \frac{x}{x^{2} + 1} = - \frac{x}{x^{2} + 1}$ , the function is odd.

The function is odd

Step 4