Operations with Functions

Other Methods of Finding Inverses

Finding the Inverse of a Function by Isolating f ^-1(x)

There is another method we can use to find the inverse of a function. Taking the inverse "reverses" x and f (x) . Thus, in the original function, substitute " x " for " f (x) " and substitute " f ^-1(x) " for " x ". Then, solve for f ^-1(x) using inverse operations in the usual manner.

Example 1: f (x) = .

1. Substitute: x =
2. Solve for f ^-1(x) :
   

   
   

   5x	=	2(f -1(x)) - 1
   5x + 1	=	2(f -1(x))
   	=	f -1(x)
   f -1(x)	=

   
   
   
3. Check: f ^-1(f (x)) = f ^-1() = = = = x

Example 2: f (x) = , x≠1 , f (x) > 0 .

1. Substitute: x =
2. Solve for f ^-1(x) :
   

   
   

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

   
   
3. Check: f ^-1(f (x)) = f ^-1( = +1 = + 1 = (x - 1) + 1 = x .

Domain of f ^-1 : x > 0 . Range of f ^-1 : f ^-1(x)≠1 .

Finding the Inverse of a Function by Graphing

We can also find the inverse of a function by graphing. The inverse of a function is a reflection of that function over the line y = x . In other words, all points (x, y) = (a, b) become (x, y) = (b, a) . The x and y coordinates of each point switch:

To find the inverse of a function, reflect the function over the line y = x . Or, find several points on the graph of y = f (x) , switch their x and y coordinates, and graph the resulting points. Connect these points with a line or curve that mirrors the line or curve of the original function.

Example: Find the inverse y = 2x - 1 by graphing: