Give a counterexample to show that W is not a subspace of C[-1,1]

Question

I am having some trouble finding a counter example for the below. IF anyone could help me out by providing one I would greatly appreciate it.

This question comes directly from the textbook Linear Algebra Gateway to Mathematics by Robert Messer.

Question: Let $W=\{f∊C[-1,1] | f(-1)=0 \lor f(1)=0\}.$ Give a counter example to show that $W$ is not a subspace of $C[-1,1].$

I suspect this fails under scalar multiplication but I'm having a hard time creating a counterexample.

Thanks again for any help.

Answer 1

$$f_1(x)=1+x\\ f_2(x)=1-x\\ f_1+f_2\notin W$$

Actually, $W$ is closed under scalar multiplication.

Answer 2

Well, this notation is horribly tortured.

Is $f$ in $W$ if and only if both equations $f(-1) = 0$ AND $f(1)=0$ hold? That's how it looks from how it was written. In which case then YES, $W$ is indeed a subspace.

Or is $W$ the set generated by (i.e., linear combinations of) the set

$\{f \in C[-1,1]$ such that $f(-1) = 0\}$ $\cup$ $\{g \in C[-1,1]$ such that $g(1) = 0\}$;

i.e., the set of functions $f$ such that EITHER $f(-1) = 0$ OR $f(1)=0$ (maybe both but not necessarily)? In which case NO, $W$ is not a subspace. Let $f(x) = x+1$, $g(x) = 1-x$, then $f+g = 2$, nonzero everywhere