Functions behave exactly as one would expect with regard to the four basic
operations of algebra (addition, subtraction, multiplication, and division).
When functions are combined by these operations, though, the domain of the
new combined function is only the elements that were shared by the domains of
the original functions. Below are the formulas for combining functions.
- The sum of two functions, f and g: (f + g)(x) = f (x) + g(x).
- The difference of two functions f and g: (f - g)(x) = f (x) - g(x).
- The product of two functions f and g: (fg)(x) = f (x)×g(x).
- The quotient of two functions f and g: ()(x) = . If g(x) = 0, the quotient is undefined.
There is one more way that functions can be combined. The fifth operation is
called the composition of two functions. The composition of the functions
f (x) and g(x) is symbolized this way: (fog)(x). It is equivalent
to f (g(x)). It is read "f of g of x." The concept is simple. First,
the value of g at x is taken, and then the value of f at that value is
taken. Let's try an example to clear things up.
Let f (x) = 3xandg(x) = x + 3. What is (fog)(x)? (fog)(x) = f (g(x)) = 3(x + 3) = 3x + 9. (gof )(x) = g(f (x)) = 3x + 3.
Now suppose that f is a one-to-one function, so that its inverse f^{-1} is a function. What is fof^{-1}? Of course, (fof^{-1})(x) = f (f^{-1}(x)). Now, f^{-1}(x) is defined to be the unique element that f
takes to x. It follows immediately that f (f^{-1}(x)) = x. Similarly, f^{-1}(f (x)) = x. Thus fof^{-1} and f^{-1}of are both the identity
function; that is, the function that takes any value of the independent
variable x to itself.