Question

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

85. \(f\left( x \right) = {\bf{1}} + {\bf{3}}{x^{\bf{2}}} - {x^{\bf{4}}}\)

Short answer

The function f is an even 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 \(f\left( x \right) = 1 + 3{x^2} - {x^4}\)to obtain\(f\left( { - x} \right)\)as shown below:

\(\begin{aligned}f\left( { - x} \right) &= 1 + 3{\left( { - x} \right)^2} - {\left( { - x} \right)^4}\\f\left( { - x} \right) &= 1 + 3{x^2} - {x^4}\\f\left( { - x} \right) &= f\left( x \right)\end{aligned}\)

Since \(f\left( { - x} \right) = f\left( x \right)\), so \(f\left( x \right)\) is an even 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\(f\left( x \right) = 1 + 3{x^2} - {x^4}\)by using the graphing calculator as shown below:

1. Open the graphing calculator. Select the “STAT PLOT” and enter the equation\(1 + 3{X^2} - {X^4}\)in the\({Y_1}\)tab.
2. Enter the “GRAPH” button in the graphing calculator.

Visualization of graph of the function\(f\left( x \right) = 1 + 3{x^2} - {x^4}\)is shown below:

It is observed that graph of the function is symmetric about \(x\)-axis, thus the function is an even function.