Exercise 41 Page 217

Before we begin, let's recall two important definitions.

• A function $f$ is an even function when $f(\text{-}x)=f(x)$ for all $x$ in its domain.
• A function $f$ is an odd function when $f(\text{-}x)=\text{-}f(x)$ for all $x$ in its domain.

Let's see how the graphs of these types of functions look. Consider the graphs above. We can see that for even functions, if $(x,y)$ is on the graph, then $(\text{-}x,y)$ is also on the graph. Meanwhile, for an odd function, if $(x,y)$ is on the graph, then $(\text{-}x,\text{-}y)$ is also on the graph. Now, consider the given function. Let's calculate $f(\text{-}x).$ Next, let's calculate $\text{-}f(x).$ Finally, let's think about what these results tell us.

$f(x)$	$f(\text{-}x)$	$\text{-}f(x)$
$x^4+3x^2-2$	$x^4+3x^2-2$	$\text{-}{x}^4-3x^2+2$

We can see above that $f(x)=f(\text{-}x).$ Therefore, $f$ is an even function.