Single Variable Calculus, Chapter 1, Review Exercises, Section Review Exercises, Problem 17

State whether the function is even, odd or neither even nor odd.



$
\begin{equation}
\begin{aligned}

\text{ a)} f(x) =& 2x^5 - 3x^2 + 2\\
f(-x) =& 2 (-x)^5 - 3 (-x)^2 + 2\\
f(-x) =& -2x^5 - 3x^2 + 2\\
\end{aligned}
\end{equation}
$


.: The function is neither odd nor even


$
\begin{equation}
\begin{aligned}

\text{b) } f(x) =& x^3 - x^7\\
f(-x) =& (-x)^3 - (-x)^7\\
f(-x) =& -x^3 + x^7\\
f(-x) =& -(x^3 - x^7)

\end{aligned}
\end{equation}
$


.: The function is odd


$
\begin{equation}
\begin{aligned}

\text{c)} f(x) =& \cos (x^2)\\
f(-x) =& \cos ((-x)^2)\\
f(-x) =& \cos (x^2)

\end{aligned}
\end{equation}
$


.: The function is even


$
\begin{equation}
\begin{aligned}

\text{d)} f(x) =& 1 + \sin x\\
f(-x) =& 1 + \sin (-x)\\
f(-x) =& 1 - \sin x

\end{aligned}
\end{equation}
$


.: The function is neither odd nor even