Is the sinc function only valid at t=0

I know the normal sinc(t) function is centered at 0.

What if I take sinc(t-2)*impulse(t) ?

In this problem the impulse is only valid at t=0 and the sinc is valid at 2....

Sinc function is usually valid for all values of t not just one value. However, according to your formula, in will have non-zero value at t=0 and zero elsewhere, and this non-zero value will equal:

sinc(-2) = sin(2)/2 = 0.01745