Suppose f(x) = x2, how does the range of g(x) = f(x) + k compare to the range of f(x)? where k is any constant value.

First:  Please use the symbol " ^ " to indicate exponentiation.  Your f(x) should be x^2, not x2.  "x^2" reads "the square of x."

Let y equal g(x) = f(x) + k:  y = g(x) = f(x) + k.  See that "k" is a y value?
If k=1, we merely translate the graph of f(x)=x^2 UP one scale division to obtain the graph of g(x) = f(x) + 1.

So, how does the range of g(x) compare to the range of f(x)?