Question

Determine whether f is even, odd, or neither. You may wish to use a graphing calculator or computer to check your answer visually.

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

Short answer

The function f is an odd function.

Step-by-step solution

Condition for even and odd function

The function \(f\) is said to be an even function if it satisfies the condition \(f\left( { - x} \right) = f\left( x \right)\). The function is an odd function if it satisfies the condition \(f\left( { - x} \right) = - f\left( x \right)\).

Check even or odd function

Replace \(x\) with \( - x\) in the function \(f\left( x \right) = \frac{x}{{{x^2} + 1}}\)to obtain \(f\left( { - x} \right)\)as shown below:

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

Since \(f\left( { - x} \right) = - f\left( x \right)\), so \(f\left( x \right)\) is an odd function.

Check the answer visually

The procedure to draw the graph of the above equation by using the graphing calculator is as follows:

To check the answer visually draws the graph of the function\(f\left( x \right) = \frac{x}{{{x^2} + 1}}\)by using the graphing calculator as shown below:

1. Open the graphing calculator. Select the “STAT PLOT” and enter the equation\(X/\left( {{X^2} + 1} \right)\)in the\({Y_1}\)tab.
2. Enter the “GRAPH” button in the graphing calculator.

Visualization of graph of the function \(f\left( x \right) = \frac{x}{{{x^2} + 1}}\) is shown below:

It is observed that graph of the function is symmetric about the origin, thus the function is an odd function