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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

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