Step 1 :Find the value of g(4) by substituting 4 into the function g(x) = 4x^2 + 7. This gives g(4) = 4*(4^2) + 7 = 71.
Step 2 :Substitute the result from step 1 into the function f(x) = 9x. This gives f(g(4)) = 9*71 = 639.
Step 3 :\( (f \circ g)(4) = \boxed{639} \)