For any integer $k,$ define $f(k)=\{\begin{array}{ccc}3k+7& ifkis\text{o}\text{d}\text{d}& \\ 5k^2& ifkis\text{e}\text{v}\text{e}\text{n}& \end{array}.$ Prove $($using fundamental definitions$)$ that for any integer $k$ that $f(k)$ is even. $[$Hint: Divide into cases depending on whether $k$ is even or odd.$]$