Section 5 - Inverse Functions


1.5 Inverse Functions

 

 -A function is represented by a set of ordered pairs. For example, the function eq=f(x)=x+1 from set A= {eq=1,2,3,4} to set B= {eq=2,3,4,5} can be written like:

 

eq=f(x)=x+1: {eq=(1,2) (2,3) (3,4) (4,5) }

 

-An inverse function, denoted by eq=f^{-1}, is formed when you switch the places of the ordered pairs. It is a function from set B to set A and is written like:

 

 

eq=f^{-1}(x)=x-1: {eq=(2,1) (3,2) (4,3) (5,4)}

 

 

*Remember: the domain of eq=f^{-1} is equal to the range of eq=f, and the range of eq=f^{-1} is equal to the domain of eq=f

 

 

-Also, a function does not always have an inverse. For a function to have an inverse, the function must be one-to-one, meaning that no two numbers in the domain of eq=f correspond to the same numbers in the range of eq=f. When looking at a graph of a function it's really easy to tell if a function is one-to-one. If the function is one-to-one, it must pass the Horizontal Line Test. If a horizontal line can pass through two points on the graph, then that function does not have an inverse.

 

 

Finding the Inverse of a Function:

 

  1. Using the Horizontal Line Test, determine whether eq=f has an inverse                                         
  2. Replace eq=f(x) with eq=y, they mean the same thing and it's easier if you do
  3. Switch the places of eq=x and eq=y, then solve for eq=y
  4. Replace eq=y with eq=f^{-1}(x)

 

 

Note: eq=f and eq=f^{-1} are only inveres of each if eq=f^{-1}(f(x))=x and eq=f(f^{-1}(x))=X

 

 

Verify that Functions are Inverses: f(g(x))=x and g(f(x)=x

 

Ex. f(x)= eq=2x^3 -1        g(x)=eq=\sqrt[3]{(x+1)/2}

 

 f(g(x))=2(eq=\sqrt[3]{(x+1)/2}eq=)^3 -1

 

              = 2((x+1)/2)-1 

               = X 

 

The twos cancel and so do the ones. One two is on top of the fraction, the other on the bottom,and     one of the ones is positive, while the other is negative-- Leaving the answer as X.

 

g(f(x))= eq=\sqrt[3]{((2x-1)+1)/2} 

             = x

The same goes for this equation, the twos and the ones cancel, making it equal to x also.

 

*Because both equations equal X they are inverses of eachother.

 

 

 

The Graph of an Inverse Function

 

 - Every Inverse is a reflection over the line y=x