Chapter 1.2, Problem 57E

To determine

To check: Whether the given statement is true or false.

Answer to Problem 57E

The given statement is false.

Explanation of Solution

Given information: The given statement is “The function $f\left(x\right)=x^4+x^2+x$ is an even function”.

As the function $f\left(x\right)=x^4+x^2+x$ is sum of even and odd powers of $x$ . Check the property $f\left(-x\right)=f\left(x\right)$ for an even function.

Substitute $-x$ for $x$ in the given function.

  $\begin{array}{c}f\left(-x\right)={\left(-x\right)}^4+{\left(-x\right)}^2+\left(-x\right)\\ =x^4+x^2-x\\ \ne f\left(x\right)\end{array}$

The function $f\left(x\right)=x^4+x^2+x$ is not an even function

Therefore, the given statement is false.