1.8. Combinations of Functions
There are several important ways in which functions may be combined. The basic arithmetic operations (addition, subtraction, multiplication, and division) may be applied to any two functions in the obvious way. For instance, for the given functions f(x) and g(x), the sum of f and g is the function h(x) defined by h(x) = f(x)+g(x) for all x. Sometimes we simply write h = f+g. It should be noted that dividing two functions f and g in general restricts the domain of the quotient, because the latter is not defined at values of x for which g(x) = 0.
Another important operation of functions is composition. Given two functions f and g, we may form new functions f◊g and g◊f defined by f◊g(x) = f(g(x)) and g◊f(x) = g(f(x)). In general, f◊g and g◊f are different, though there are important exceptions.
Example 1: Consider the functions f(x) = 2x and g(x) = x2+5. What are f◊g(x) and g◊f(x)?
Solution: We compute f◊g(x) = f(g(x)) = 2g(x) = 2(x2+5) = 2x2 +10 and g◊f(x) = g(f(x)) = f(x)2+5 = (2x)2+5 = 4x2+5. Note that these functions are not equal.