If the inverses of two functions are both functions, will the inverse of the sum or difference of the original functions also be a function?

No. Consider the functions given below :

Let f and g be two functions from the set of real numbers to the set of real numbers defined by

f(x) = x , g(x) = x and h(x) = - x

Clearly, inverses of these functions are functions.

But, f(x) + h(x) = x + ( - x) = 0 does not have inverse.

Also, f(x) - g(x) = 0 does not have inverse.

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