Evaluate f(-x)=x/(x^2+1)

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

Step 1

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

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

Step 2

Simplify the left side.

Dividing two negative values results in a positive value.

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

Cancel the common factor of $x$ .

Cancel the common factor.

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

Divide $f$ by $1$ .

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

Step 3

Simplify the right side.

Multiply the numerator by the reciprocal of the denominator.

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

Cancel the common factor of $x$ .

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

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

Cancel the common factor.

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

Rewrite the expression.

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

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

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

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

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

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

Move the negative in front of the fraction.

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