Here are a few cases and I'll explain of g(x) differs from f(x):

g(x) = f(x-a)

f(x) is shifted to the right by a to get g(x)

g(x) = f(x+a)

f(x) is shifted to the left by a to get g(x)

g(x) = f(x) + a

f(x) is shifted up by a to get g(x)

g(x) = f(x) - a

f(x) is shifted down by a to get g(x)

g(x) = -f(x)

f(x) is flipped along the x-axis to get g(x)

I hope this helps and is what you wanted. This is all that can be said generally. If f(x) has a specific form (like a sine wave for example) then more things can be said.