   SEARCH HOME Math Central Quandaries & Queries  Question from jess, a student: if g(x)=2x+3 find g^-1(g(4)) how do I solve I can't figure out what to do with the 4 Hi Jess.

Short solution:
g-1(g(x)) means to feed x into the g(x) formula, then take that answer and feed it backwards through it again !

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).

Long solution:
g(x) = 2x + 3
so g(4) = 2(4) + 3 = 8 + 3 = 11.

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):
So
g(x) = 2x + 3
becomes
x = 2(g-1(x)) + 3

Next I solve for g-1(x):
x - 3 = 2(g-1(x))
g-1(x) = (x - 3) / 2

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:
x = ( h-1(x) )2
± sqrt(x) = h-1(x)
h-1(x) = ± sqrt(x).

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.

Cheers,
Stephen La Rocque.     Math Central is supported by the University of Regina and The Pacific Institute for the Mathematical Sciences.