Chapter 3, Problem 1RE

To determine

Whether the statement “$f\left(-x\right)=-f\left(x\right)$ for any function $f\left(x\right)$” is true or false.

Answer to Problem 1RE

The given statement is $\underset{\_}{\text{false}}$.

Explanation of Solution

Calculation:

The given statement is $f\left(-x\right)=-f\left(x\right)$ for any function $f\left(x\right)$”.

Consider the function $f\left(x\right)=x^4$.

Replace x by –x in $f\left(x\right)=x^4$.

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

The $f\left(-x\right)=-f\left(x\right)$ is false for the function $f\left(x\right)=x^4$.

Hence, the given statement is $\underset{\_}{\text{false}}$.