Composition and Inverses

1. Composition of Functions
   

   Example:  
   
   Sociologists in Holland determine that the number of people y waiting in a water ride at an amusement park is given by

           y = 1/50C²  + C + 2

   where C is the temperature in degrees C.  The formula to convert Fahrenheit to Celsius C is given by

           C = 5/9 F + 160/9

   To get a function of F we compose the two function:

           y(C(F)) = (1/50)[5/9F + 160/9]² + (5/9F + 160/9) + 2
   
   

   Exercises: 

    If 
   
           f(x) = 3x + 2  
   
           g(x) = 2x² + 1 
   
           h(x) =
   
           c(x) = 4
   
   

   1.  Find f(g(x))
      

   2. Find f(h(x))
      

   3. Find f(f(x))
      

   4. Find h(c(x))
      

   5. c(f(g(h(x))))

   
   
   

2. 1-1 Functions
   

   Definition A function f(x) is 1-1 if            f(a) = f(b)  implies that            a = b

   
   
   Example:  
   
   If 
   
           f(x) = 3x + 1 
   
   then 
   
           3a + 1 = 3b + 1 
   
   implies that 
   
           3a = 3b

   hence 
   
           a = b 
   
   therefore f(x) is 1-1.
   
   

   Example:  
   
   If 
   
           f(x) = x²  
   
   then 
   
           a² = b²  
   
   implies that 
   
           a² - b²  = 0 
   
   or that 
   
           (a - b)(a + b) = 0 
   
   hence

           a = b or a = -b
   
   For example 
   
           f (2) = f (-2) = 4
   
   Hence f (x) is not 1-1.

   
   
   

3. Horizontal Line Test

   If every horizontal line passes through f(x) at most once then f(x) is 1-1.

   
   
   
   

4. Inverse Functions
   
   
   

   Definition     A function g(x) is an inverse of f (x) if           f (g(x)) = g(f (x)) = x

   
   
   Example:  
   
   The volume of a lake is modeled by the equation

           V(t) = 1/125 h³

    Show that the inverse is

           h(N) = 5V^1/3  

   We have 
   
           h(V(h)) = 5(1/125h³)^1/3 = 5/5h = h

   and 
   
           v(h(V)) = 1/125(5V^1/3)³ = 1/125(125V) = V

   
   
   

5. Step by Step Process for Finding the Inverse:
   
   

   1. Interchange the variables
      

   2. Solve for y
      

   3. Write in terms of f ^-1(x)

   
   
   Example:
   
   Find the inverse of 

           f (x) = y = 3x³ - 5
   
   

   1. x =  3y³ - 5
      

   2. x - 5 =  3y³ , (x - 5)/3 =  y³ , [(x - 5)/3]^1/3 
      

   3. f ^-1(x) =  [(x - 5)/3]^1/3 

   
   
   

6. Graphing:  

   To graph an inverse we draw the y = x line and reflect the graph across this line.

   To interactively view the graph of an inverse click here:
   
   http://mathcsjava.emporia.edu/~greenlar/Inverse/inverse.html

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