f(x) and g(y) are functions of x and y, respectively, and f (x) = g (y) for all real values of x and y. Which of the following option is necessarily TRUE for all conditions?

Let’s analyze from the given information:
We know that f(x) and g(y) are functions of x and y, respectively, and f(x) = g(y) for all real values of x and y.
Since f(x) and g(y) are equal for all real values of x and y, this means that both functions are constant.
Now let’s consider the options:
(A) f(x) = 0 and g(y) = 0: This is not necessarily true because the functions can be any constant value, not necessarily zero.
(B) f(x) = g(y) = constant: This is correct. We already established that both functions are constant, so they are equal to some constant value.
(C) f(x) ≠ constant and g(y) ≠ constant: This is not true, as both functions are constant.