Question

Composite Functions.Given the functions \(g(x)=x-4\) and \(f(x)=x^{2},\) write the composite function \(f[g(x)]\).

Short answer

The composite function is: \(f[g(x)]=f(x-4)=(x-4)^{2}\)

Step-by-step solution

Identify the Outer and Inner Functions

First, determine which function is the outer function and which one is the inner function in the composition process. In the composite function notation, the function that is applied first is the inner function and the function that is applied second is the outer function. In this case, the inner function is g(x) and the outer function is f(x).

Evaluate the Inner Function

Apply the inner function g(x) to x. That is, compute g(x) = x - 4.

Substitute into Outer Function

Replace the x in the outer function f(x) = x^{2} with the result from the inner function. That is, wherever there is an x in f(x), substitute it with the expression for g(x), which is (x - 4).

Key concepts

Understanding Inner and Outer Functions

When it comes to the concept of composite functions, two important components are the inner and the outer functions. The inner function is the one that you apply first. Think of it as the initial step in a two-part process. Applying this to a value gets you ready for the next part. The outer function, on the other hand, is the second step that uses the result of the inner function as its input.

In the given example, the inner function is denoted as g(x) which equals to x minus 4. To use this in a composite function, one would first perform this subtraction before anything else. The outer function in the problem, f(x), stands for taking the square of its input. Therefore, after determining the result of the inner function g(x), we use that result as the input for f(x).

Navigating Function Composition

Function composition involves combining two or more functions in a specific order to create a new function. The notation f[g(x)] indicates that g(x) is to be evaluated first (inner function), and then the resulting value is to be used as the input for f(x) (outer function).

Imagine function composition as a machine with two compartments: the first compartment is labeled 'g' and the second 'f'. You drop your x into the 'g' compartment, it does its job and hands off the output to the 'f' compartment to complete the process. The final result is the outcome of this combined operation. In our exercise, after evaluating g(x) = x - 4, we place the output into f(x) = x^2 to get the composite function f(g(x)) = (x-4)^2.

Evaluating Functions Step by Step

Evaluating functions is much like following a cooking recipe. Each step needs to be executed in the correct sequence to yield the desired result. As illustrated in the problem, the first step is to determine and compute the inner function, which in this case is g(x) = x - 4. This evaluation is akin to measuring out your ingredients before mixing them.

Once the inner function has been calculated, you then substitute this result into each instance of x within the outer function, f(x) = x^2, just like combining ingredients in a recipe. By substituting g(x) into f(x), you effectively blend these functions together to arrive at the final expression for the composite function: f[g(x)] = (x - 4)^2. This step-by-step approach ensures clear and precise execution of function composition.