Section 4-2:  Operations on Functions
 
 
Operations on functions are similar to operations on numbers.  Each operation has a formal definition which produces a new function!!
  
 
       Operations on Functions 
 
Each function is defined for all x in the domains of both f and g.
 
   1)  Sum of f and g:        (f + g)(x) = f(x) + g(x)
 
   2)  Difference of f and g:            (f - g)(x) = f(x) - g(x)
 
   3)  Product of f and g:            (f . g)(x) = f(x) . g(x)
 
   4)  Quotient of f and g:        (f/g)(x) = f(x)/g(x), g(x) not equal to 0
 

 
Sample Problems
 
Let f(x) = x + 3 and g(x) = x - 2
 
1)  The sum = (f + g)(x) = f(x) + g(x) = x + 3 + x - 2 = 2x + 1
 
2)  The difference = (f - g)(x) = f(x) - g(x) = (x + 3) - (x - 2) = 5
 
3)  The product = (f . g)(x) = f(x) . g(x) = (x + 3)(x - 2) = x2 + x - 6
 
4)  The quotient = (f/g)(x) = f(x)/g(x) =  (x + 3)/(x - 2), x does not = 2

 
Another way of combining functions is the composite function.  The composite function is denoted by:
and is = f(g(x)).  x must be in the domain of g(x) and g(x) must be in the domain of f(x).
 
Example 1
 
Find f(g(x)) and g(f(x)) for the functions f(x) = x + 2 and g(x) = x2
 
             f(g(x)) = f(x2) = x2 + 2.  The function g(x) squares any number.  The function f(x) simply adds two to any number.  G(x) is applied first (it squares x) and then f(x) is applied (it adds two).  The domain is all real numbers.
 
             g(f(x)) = g(x + 2) = (x + 2)2   This time, f(x) is applied first and then g(x).  Take any number and add two first then square the result.  The domain is all real numbers.  Notice that the answers are not the same illustrating that the composite function is not commutative.  This means the order in which the problem is written is important.

Example two
 
Find f(g(x)) and g(f(x)) for the functions f(x) = 1/x and g(x) = 2x.
 
             f(g(x)) = f(2x) = 1/(2x).  The domain is all real numbers but zero.
 
             g(f(x)) = g(1/x) = 2/x.  The domain is again all real numbers but zero.

 
Example three
 
Find f(g(x)) and g(f(x)) for the functions f(x) = x + 3 and g(x) = 2/x
 
             f(g(x)) = f(2/x) = 2/x + 3.  The domain is all real numbers but 0.
 
             g(f(x)) = g(x + 3) = 2/(x + 3).  The domain is all real numbers but -3.

Sample Problems
 
1)  On the same axes graph f(x) = x2, g(x) = x2 + 3, and h(x) = x2 - 2
 
The effect of adding three, simply moved the graph up 3 units while the effect of subtracting two simply moved the graph down two units.
 
2)  Let f(x) = 4x, g(x) = 3x + 2 and h(x) = x2
 
         Find f(g(h(3)))
                                 = f(g(9)) = f(29) = 116
 
         Find h(g(f(-1)))
                                     = h(g(-4)) = h(-10) = 100
 
         Find h(h(h(2)))
                                  = h(h(4)) = h(16) = 256
 
         Find h(g(f(x)))
                                 = h(g(4x)) = h(12x + 2) = (12x + 2)2

 
Let's head over to the next section! 
I think I need to head back!