Even and Odd Functions

Problem 111: Determine if $f(x)$ is even or odd:

(1) $\begin{equation*} f(x) = \frac{x^4 + x^2}{x}.\end{equation*}$

Solution: Before we go ahead and solve the problem, let’s remind ourselves that a function is even if $f(-x) = f(x)$ . Moreover, a function is odd if $f(-x) = -f(x)$ . That is,

(2) $\begin{align*} f(x) & = \frac{x^4 + x^2}{x} \\ f(-x) & = \frac{(-x)^4 + (-x)^2}{(-x)} \\ & = \frac{x^4 + x^2}{-x} \\ & = - \frac{x^4+x^2}{x} \\ & = -f(x). \end{align*}$

Since $f(-x) = -f(x)$ , we say that $f(x)$ is an odd function.

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