Question

Is f of x always equal to g of x

Answer 1

Of course not. 

' f(x) ' is the description of a function,and ' g(x) ' is the description of
a function.  There are an infinite number of different possible functions,
so f(x) and g(x) are usually not the same one.

f(x) is equal to g(x) for any 'x' only when the two descriptions are the
same function.  Otherwise f(x) is equal to g(x) only at the points where
their graphs intersect.