Problem 26 - College Algebra

Question

Find functions \(f(x)\) and \(g(x)\) so the given function can be expressed as \(h(x)=f(g(x))\). $$ h(x)=(x+2)^{2} $$

Step-by-Step Solution

Answer

\(f(x) = x^2\), \( g(x) = x+2\)

Step 1: Identify the Outer Function

Look at the form of the given function \(h(x) = (x+2)^2\). Identify the operation applied last, which is squaring in this case. Therefore, the outer function is \(f(u) = u^2\), where \(u = g(x)\).

Step 2: Identify the Inner Function

The expression inside the square is \(x+2\). Thus, the inner function \(g(x)\) is the expression itself, so \(g(x) = x+2\).

Step 3: Write the Composite Function

Now, express \(h(x)\) as a composition of \(f(x)\) and \(g(x)\) using the expressions we determined. Substitute \(g(x)\) into \(f(x)\) to get \(h(x) = f(g(x)) = (g(x))^2 = (x+2)^2\). This confirms that the functions \(f(x) = x^2\) and \(g(x) = x+2\) are correctly identified.

Key Concepts

Outer Function

When we talk about function composition, the term "outer function" refers to the function that acts on the result of another function. In the solution given, the problem involves the expression \(h(x) = (x+2)^2\). Here, the operation that stands out is the squaring, which is the final operation we apply. Hence, we identify the outer function as \(f(u) = u^2\). The outer function wraps around the inner function and completes the transformation. Think of it as placing a piece of candy in a wrapper. You can imagine every function composition as a similar scenario, where outer function adds an extra layer or operation to the outcome of the previous function.

Understanding the role of the outer function is like identifying the coat you put on top of your outfit; it’s the final layer that changes or enhances the whole ensemble.

Inner Function

The "inner function" is the function that operates first, preparing the input for the outer function. In our problem, we started with \(h(x) = (x+2)^2\). We identified that inside the parentheses, \(x+2\) is the inner function. So, we can express the inner function as \(g(x) = x+2\). The inner function sets the stage for what comes next, sending its output to the outer function for further transformation.

The role of the inner function is crucial. Without it, the outer function would have nothing to operate on. It's like cleaning a window before putting a frame around it. Ensuring the inner function is correctly identified is a key step in the process of function composition.

Composite Function

A "composite function" is created when two functions are combined so that the output of one function becomes the input of another. In the given exercise, our composite function is \(h(x) = f(g(x))\). Here, \(g(x) = x+2\) is fed into \(f(u) = u^2\). This gives us \(h(x) = (g(x))^2 = (x+2)^2\), confirming that our composed function retains its original form.

Composite functions offer a powerful way to build complex calculations using simpler expressions. They allow for a flexible approach to problem-solving where functions can be nested within each other. Think of it as cooking a layered dish, where each layer adds a unique taste, yet the final outcome is a delightful combination.