Question

Determine whether even, odd, or neither. If you have a graphing calculator, use it to check your answer visually.

\({\bf{f}}\left( {\bf{x}} \right){\bf{ = }}\frac{{\bf{x}}}{{{{\bf{x}}^{\bf{2}}}{\bf{ + 1}}}}\)

Short answer

The function \(f\left( x \right) = \frac{x}{{{x^2} + 1}}\) is odd.

Step-by-step solution

Determination of even or odd for given function theoretically

The given function will even function if\({\bf{f}}\left( {{\bf{ - x}}} \right){\bf{ = f}}\left( {\bf{x}} \right)\). The given function will be odd if\({\bf{f}}\left( {{\bf{ - x}}} \right){\bf{ = - f}}\left( {\bf{x}} \right)\).

Replace x by –x for a given function to check for odd or even.

\(\begin{array}{c}f\left( { - x} \right) = \frac{{\left( { - x} \right)}}{{{{\left( { - x} \right)}^2} + 1}}\\f\left( { - x} \right) = \frac{{ - x}}{{{x^2} + 1}}\\f\left( { - x} \right) = - \left( {\frac{x}{{{x^2} + 1}}} \right)\\f\left( { - x} \right) = - f\left( x \right)\end{array}\)

The given function is odd.

Verification of answer graphically

The graph of the given function is shown below:

Since the given function is symmetric with respect to the origin. So the given function will be odd.

Therefore, the given function is odd.