Answer
$W$ is not a vector subspace of $C(-\infty, \infty)$.
Work Step by Step
Let $f(x)=x+1, g(x)=-x+2\in W$. Now, $$f(x)+ g(x)=3$$
which means that $W$ is not closed under addition. Hence, $W$ is not vector subspace of $C(-\infty, \infty)$.
You can help us out by revising, improving and updating this answer.
Update this answerAfter you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.