Question

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

\({\bf{f}}\left( {\bf{x}} \right){\bf{ = x}}\left| {\bf{x}} \right|\)

Short answer

The function \(f\left( x \right) = x\left| x \right|\) is odd.

Step-by-step solution

Determination of even or odd for the given function

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.

\(\begin{array}{c}f\left( { - x} \right) = \left( { - x} \right)\left| {\left( { - x} \right)} \right|\\f\left( { - x} \right) = - \left( {x\left| x \right|} \right)\\f\left( { - x} \right) = - f\left( x \right)\end{array}\)

The given function is odd.

Verification of answer graphically

The graph of given is shown below:

Since the given function is symmetric about the origin. So the given function will be odd.

Thus, the given function is odd.