Question

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

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

Short answer

The function \(f\left( x \right) = 1 + 3{x^2} - {x^4}\) is an even function.

Step-by-step solution

Given data

Consider the following function,

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

Determine whether the function is even, odd, or neither

To determine whether the function is even or odd, use even and odd function 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\(f\left( x \right)\).

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

The function\(f\left( { - x} \right) = f\left( x \right)\)so\(f\)is an even function.

Therefore, the function\(f\left( x \right) = 1 + 3{x^2} - {x^4}\)is an even function.

The graph is given below

Use the graphing utility to check the solution.

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

From the graph, observe that the function\(f\left( x \right) = 1 + 3{x^2} - {x^4}\)is symmetric,

about the\(y - axis\).