Sure, let's approximate the solution to the equation f(x) = g(x) where f(x) = 3x+3 and g(x) = 2^(x-3)+4 using successive approximation.
Starting with our initial guess of x = 0, we perform three iterations of our approximation process.
For the first iteration, we plug 0 into both the function f and g. This gives us:
f(0) = 3(0) + 3 = 3
g(0) = 2^(0-3) + 4 = 1/8 + 4 = 4.125
We then take an average of these two values to get our new approximation for x, which results in x = (3 + 4.125)/2 = 3.5625.
For the second iteration, we now put our new approximation into the two functions:
f(3.5625) = 3(3.5625) + 3 = 13.6875
g(3.5625) = 2^(3.5625 - 3) + 4 = 2^0.5625 + 4 = 4.421875
We average these two results to get our next approximation for x: x = (13.6875 + 4.421875)/2 = 9.0546875.
For our final iteration, we plug this new x value back into our functions:
f(9.0546875) = 3(9.0546875)+3 = 30.1640625
g(9.0546875) = 2^(9.0546875 - 3) + 4 = 101.3963274
Taking another average gives us, x = (30.1640625 + 101.3963274)/2 = 65.78019495
Therefore, after three iterations of successive approximation, the value of x that solves f(x) = g(x) to a reasonable degree of accuracy is approximately 65.78. Please remember that, depending on how many iterations one performs, the solution may become more accurate.