What is a Composite Function in Terms of Input and Output?

A composite function is essentially a function machine where the output of one function is fed directly into another function as its input. This cascading relationship defines how the final output is produced from the initial input.

Understanding Function Composition

As described in the reference, the process of combining functions so that the output of one function becomes the input of another is known as a composition of functions. The function that results from this process is called a composite function.

Think of it as a two-step (or multi-step) operation:

1. You start with an initial input value.
2. This input goes into the first function (let's call it g).
3. Function g processes this input and produces an output.
4. This output from g then becomes the input for the second function (let's call it f).
5. Function f processes this new input (which was g's output) and produces the final output of the composite function.

This entire sequence is represented by the notation f∘g(x) = f(g(x)). Here, x is the initial input to g, g(x) is the output of g (which becomes the input to f), and f(g(x)) is the final output of the composite function f∘g.

The Input-Output Flow

Let's visualize the flow:

• Initial Input: x
• First Function: g
• Intermediate Output / Second Input: g(x)
• Second Function: f
• Final Output: f(g(x))

This demonstrates how the output of the inner function (g) serves as the input for the outer function (f), forming a single, new function (f∘g).

Example

Consider two simple functions:

• g(x) = x + 2 (Adds 2 to the input)
• f(x) = x² (Squares the input)

To find the composite function f∘g(x) in terms of input and output:

1. Input: Start with an initial value, say x = 3.
2. Process by g: The input 3 goes into g(x) = x + 2.
   • g(3) = 3 + 2 = 5.
   • The output of g is 5.
3. Process by f: The output 5 from g becomes the input for f(x) = x².
   • f(5) = 5² = 25.
   • The final output of the composite function is 25.

So, for the composite function f∘g(x), when the input is 3, the output is 25. The rule for the composite function f∘g(x) is f(g(x)) = f(x + 2) = (x + 2)².

Here's a look at the relationship:

Process	Input	Function	Output	Role of Output
First Step (g)	Initial x	g	g(x)	Input for f
Second Step (f)	g(x)	f	f(g(x))	Final Result
Composite (f∘g)	Initial x	f∘g	f(g(x))	Final Result

This table clarifies how the intermediate output of the first function seamlessly transitions into the input of the second function to yield the final output of the composite function.

Understanding this input-output relationship is key to working with composite functions in various mathematical and real-world applications, such as calculating the total cost considering taxes and discounts, or modeling systems where one process's result feeds into another.