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Operations with Functions

Other Methods of Finding Inverses

Problems

Problems

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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