Composite Functions
You may have noticed that when we evaluate a function, we typically evaluate it for a a number or variable, but we can go one step further and evaluate a function of another function. This process is very similar to when you would substitute a variable with an expressions. For example:
When working with composite functions, we follow a similar process. If we are given two functions f(x) and g(x) and we want to find f(g(x)) we simply replace all instances of x in the function f(x) with whatever g(x) is set equal to. Then we simplify.
When we have a composite function, the notation f(g(x)) can be written as . This expression is read, "f of g of x" and means that we are replacing the x values of inputs of f(x) with another function, in this case g(x).
Evaluating a Composite Function
Let's take a look at how to actually evaluate a composite function. Suppose we are told that and that and asked to find f(g(x)). To do this we can take the following steps:
Sometimes we may be asked to solve a particular composite function when x is equal to a given value, for example x = 2. To make this evaluation we follow the same steps we did above then at the end we simply substitute in the value we are given for x. Let's see how we do this:
Alternatively we would have evaluated g(2) first and then plug that value in for x in the function f(x). Either method would give us the same result.
Sometimes you may be asked to find g(f(x)). In this case we do the same procedures only we replace each variable in g(x) with the expression for f(x).
Evaluating f(f(x))
Sometimes you may comes across a situation where you will need to evaluate a function for itself, namely f(f(x)). This simply means that you are plugging in the expression of the function for each x in the function. For example, if f(x) = 2x - 1 and we want to find f(f(x)) we would do the following:
Like with the previous problem, you may be asked to find the composite function of itself when x is a given value, for example f(f(0)). Just like before we simply need to replace each instance of x in the function f(f(x)) with the given value.
We can show this on the graph below, by noting that if f(x) = 2x - 1 then f(0) = -1. Plugging in -1 in for x in f(x) we get f(-1) = -3, which is the same answer we found doing this algebraically.
the combination of functions, such that the output of one function is the input of another