How to restrict the domain of a function

I think the easiest way is to use Condition which has the operator form /;.

f[x_ /; -1 <= x <= 1, c_] := -(x + c)^2

Then

With[{c = -1/4}, Plot[f[x, c], {x, -2, 2}]]

Amplifying on m_goldberg's answer: The Condition can include a Message to use when the argument is outside the allowed interval.

ClearAll[f]

f::arg = "Argument value of `1` must be in the closed interval {-1, 1}.";

f[x_, c_] /; If[TrueQ[-1 <= x <= 1], True, Message[f::arg, x]] := -(x + c)^2

Then, an argument outside the interval generates a Message

f[2, c]

The Message can be turned Off

With[{c = -1/4}, Off[f::arg]; Plot[f[x, c], {x, -2, 2}]]