Algebra Examples

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

Step 1

Divide each term in $f (- x) = \frac{x}{x^{2} + 1}$ by $- x$ .

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

Step 2

Simplify the left side.

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Step 2.1

Dividing two negative values results in a positive value.

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

Step 2.2

Cancel the common factor of $x$ .

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Step 2.2.1

Cancel the common factor.

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

Step 2.2.2

Divide $f$ by $1$ .

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

Step 3

Simplify the right side.

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Step 3.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 3.2

Cancel the common factor of $x$ .

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Step 3.2.1

Factor $x$ out of $- x$ .

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

Step 3.2.2

Cancel the common factor.

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

Step 3.2.3

Rewrite the expression.

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

Step 3.3

Multiply $\frac{1}{x^{2} + 1}$ by $\frac{1}{- 1}$ .

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

Step 3.4

Cancel the common factor of $1$ and $- 1$ .

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Step 3.4.1

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

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

Step 3.4.2

Move the negative in front of the fraction.

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