So simply, g-1(g(4)) = 4, because the g-1( ) cancels the g( ). This is valid because g(x) is defined as a one-to-one function (in this case, a line).
This means that g-1(g(4)) = g-1(11).
Now g-1(x) means to do things backwards, so I replace g(x) with x and x with g-1(x):
Next I solve for g-1(x):
So if we want g-1(11), then x = 11 for this step and we just fill it in:
g-1(11) = (11 - 3) / 2 = 8 / 2 = 4.
So the answer is 4, just like it was in the quick solution.
You have to be careful that the function you are reversing is one-to-one when using the quick solution, whereas the long solution will show you a problem.
For example, consider h(x) = x2. What is h-1(h(-4)) ?
The short answer tells us that h-1( h(x) ) = x, so the answer would be -4, right? Not so fast!
The long answer says h(-4) = (-4)^2 = 16 so h-1(h(-4)) = h-1(16). To find h-1(x) we proceed:
So with x now equal to 16, we have h-1(16) = ± 4 rather than -4. So we'd have an incomplete answer. The real answer to the question is what this long solution method gave us!
This is because h(x) = x2 is not a one-to-one function - there are two different x values that generate the same result, 16.
So use the quick way if you are sure (and can satisfy yourself and your teacher) that it is a one-to-one function (all line equations are one-to-one functions, for example), but if in doubt, use the long way.
Math Central is supported by the University of Regina and The Pacific Institute for the Mathematical Sciences.