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Consider the operator \( \hat{A} \) such that for function \( f(x) \) we have:

\[ \hat{A}f(x) = f(x+a) + f(x-a) \].

The domain for all functions is on \( (-\infty, \infty) \).

(a) What are the two conditions that an operator on functions must satisfy to be a \emph{linear} operator?

(b) Prove \( \hat{A} \) is a linear operator.

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