Search Menu

Contents

Other Methods of Finding Inverses

Other Methods of Finding Inverses

Other Methods of Finding Inverses

Other Methods of Finding Inverses

Other Methods of Finding Inverses

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:

Inverse of a Graph

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:

Inverse of y = 2x - 1