On-line Math 21

Inverse Functions

Example 1 If

f(x) = 2x+1 x-1 ,

find the formula for the inverse function g = f^-1 .

Solution

Start out with the equation y = f(x) , and solve that equation for x . It will automatically be the formula for the inverse, x = f^-1(y) . But, of course, now you are supposed to think of the inverse as just another function, so you write it, like all generic functions, as a function of an x , y = f^-1(x) , That last bit is accomplished after you first solve the original equation for x , x = f^-1(y) . Then just switch the roles of x and y .

Here's how it works:

y = f(x) = 2x+1 x-1 .

Then, solve for x :

y = 2x+1 x-1 y(x-1) = 2x+1 yx-y = 2x+1. Now, put all x¢s on the left: yx-2x = y+1 x(y-2) = y+1, or x = y+1 y-2 .

Then, this says that

f-1(y) = y+1 y-2 .

To take the last step, meaningless though it may be,

f-1(x) = x+1 x-2

is the formula for the inverse function.

You can check to be sure that this is the inverse, since

f(f-1(x)) = f æ ç è x+1 x-2 ö ÷ ø = 2 æ ç è x+1 x-2 ö ÷ ø +1 æ ç è x+1 x-2 ö ÷ ø -1 = 2(x+1)+1(x-2) (x+1)-(x-2) , (multiplying top and bottom by (x-2)), = 2x+2+x-2 x+1-x+2 = 3x 3 = x,

which is what we wanted to show. You can also verify that

f-1(f(x)) = x

as well, but I'll leave that to you.

Copyright (c) 2000 by David L. Johnson.