Question

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

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

Short answer

The function f is neither even nor 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 anodd 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 \(f\left( x \right) = 1 + 3{x^3} - {x^5}\)to obtain\(f\left( { - x} \right)\)as shown below:

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

Since \(f\left( { - x} \right) \ne f\left( x \right)\), and \(f\left( { - x} \right) \ne - f\left( x \right)\), so \(f\left( x \right)\) is neither even nor 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 draw the graph of the function\(f\left( x \right) = 1 + 3{x^3} - {x^5}\)by using the graphing calculator as shown below:

1. Open the graphing calculator. Select the “STAT PLOT” and enter the equation\(1 + 3{X^3} - {X^5}\)in the\({Y_1}\)tab.
2. Enter the “GRAPH” button in the graphing calculator.

Visualization of graph of the function \(f\left( x \right) = 1 + 3{x^3} - {x^5}\) is shown below:

It is observed that graph of the function is neither symmetric about \(x\)-axis nor symmetric about origin. Thus, it is neither even nor odd function.