Let's eliminate answers choices by plugging easy-to-work-with numbers into each function.

For example, the table tells us that f(0) = 1, so let's plug x = 0 into each function and eliminate those that don't evaluate to 1.
We get:
(A) $f(0) = 0^2 + 0 - 1 =-1$. Eliminate
(B) $f(0) = 0^2 - 1 = -1$. Eliminate
(C) $f(0) = -(0^2) + 0 + 1 = 1$. Keep
(D) $f(0) = -(0^2) + 1 = 1$. Keep
(E) $f(0) = 0^2 - 0 + 1 = 1$. Keep

The table also tells us that, when f(1) = 0, so let's plug x = 1 into each remaining function and eliminate those that don't evaluate to 0.
We get:
(C) $f(1) = -(1^2) + 1 + 1 = 1$. Eliminate
(D) $f(1) = -(1^2) + 1 = 0$. Keep
(E) $f(1) = 1^2 - 1 + 1 = 1$. Eliminate

By the process of elimination, the correct answer is D