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-1of are both the identity function; that is, the function that takes any value of the independent variable x to itself.