Question

Determine whether\({\bf{f}}\)is even, odd, or neither. If you have a graphing calculator, use it to check your answer visually.

64.\({\bf{f}}\left( {\bf{x}} \right){\bf{ = 1 + 3}}{{\bf{x}}^{\bf{3}}}{\bf{ - }}{{\bf{x}}^{\bf{5}}}\)

Short answer

The function \(f\left( x \right) = 1 + 3{x^3} - {x^5}\) is neither even nor odd.

Step-by-step solution

Given data

Consider the following function,

\(f\left( x \right) = 1 + 3{x^3} - {x^5}\)

Check whether the function is even, odd, or neither

To determine whether the function is even or odd, use the definitions.

If a function\(f\)satisfies\(f\left( { - x} \right) = f\left( x \right)\)for every number\(x\)in its domain,

then\(f\)is called an even function.

If a function\(f\)satisfies\(f\left( { - x} \right) = - f\left( x \right)\)for every number\(x\)in its domain,

then\(f\)is called an odd function.

Replace\(x\)with\( - x\), in\(f\left( x \right)\).

\(\begin{array}{l}f\left( { - x} \right) = 1 + 3{\left( { - x} \right)^3} - {\left( { - x} \right)^5}\\\;\;\;\;\;\,\,\,\,\, = 1 - 3{x^3} + {x^5}\end{array}\)

The function is neither\(f\left( { - x} \right) \ne f\left( x \right)\)nor\(f\left( { - x} \right) \ne - f\left( x \right)\), so the given functionis neither even nor odd.

Therefore, the function\(f\left( x \right) = 1 + 3{x^3} - {x^5}\)is neither even nor odd.

The graph is given below

Use the graphing utility to check the solution.

Sketch the graph of\(f\left( x \right)\).

From the graph, it can be observed that the function \(f\left( x \right) = 1 + 3{x^3} - {x^5}\) is neither symmetric about the \(y - axis\) nor about the origin.