Problem 52 - Algebra and Trigonometry

Question

Express the function in the form \(f \circ g\) $$ Q(x)=\frac{1}{x+3} $$

Step-by-Step Solution

Answer

The function is \(f(g(x)) = \frac{1}{x+3}\) with \(f(u) = \frac{1}{u}\) and \(g(x) = x+3\).

Step 1: Identify the Inner Function

We need to express the given function in the form \(f(g(x))\). Start by identifying what part of the function can act as \(g(x)\). In this case, the expression \(x+3\) can be considered as \(g(x)\). Thus, we define \(g(x)=x+3\).

Step 2: Determine the Outer Function

Now, using \(g(x) = x + 3\), we look at what operation is done to \(g(x)\) to get \(Q(x)\). We have \(Q(x) = \frac{1}{g(x)}\). So, the outer function forms the reciprocal of the input, hence \(f(u) = \frac{1}{u}\) where \(u = g(x)\).

Step 3: Combine the Functions

Combine the functions \(f(u)\) and \(g(x)\) to express \(Q(x)\) in terms of \(f(g(x))\). Therefore, \(Q(x) = f(g(x)) = \frac{1}{g(x)} = \frac{1}{x+3}\).

Step 4: Verify the Composition

It's always good to verify if the composition indeed results in the original function. Substitute \(g(x) = x+3\) into \(f(u) = \frac{1}{u}\): \(f(g(x)) = f(x+3) = \frac{1}{x + 3}\), which matches \(Q(x)\).

Key Concepts

Inner Function

In function composition, we start by identifying what's called the "inner function." This is the function that gets applied or operated upon first inside a composition of two functions. Think of it as the core building block that the entire function depends on. For the exercise given, we're looking at the expression \(Q(x) = \frac{1}{x+3}\).
We need to recognize which part of this expression can be our inner function. Here, the expression \(x+3\) is identified as part of the calculation that happens first before anything else. Thus, we define this as the inner function \(g(x) = x+3\).
Breaking down a function into simpler parts makes it easier to work with complex expressions, reinforcing the fundamental understanding of function composition.

Outer Function

Once we've identified the inner function, the next step is to determine the "outer function," which is applied to the result of the inner function. In simple terms, it's the function that "wraps around" the inner function to fully articulate the original function expression. In this exercise, after recognizing \(g(x) = x+3\) as the inner function, we need to see what's done next to this result to achieve \(Q(x)\).
Here, we are taking the reciprocal of \(g(x)\), meaning we are finding \(\frac{1}{g(x)}\). This makes our outer function \(f(u) = \frac{1}{u}\), where \(u\) is just a placeholder that stands in for whatever input \(g(x)\) provides, which is \(x+3\).
Understanding the role of the outer function helps consolidate the pieces into a coherent expression that aligns with the original function.

Function Notation

Function notation is the language we use to describe functions and their inputs and outputs compactly and clearly. It is incredibly useful in expressing compositions like \(f(g(x))\) because it highlights the order in which operations are performed.
In the context of the exercise, using function notation, we wrote \(g(x) = x + 3\) and \(f(u) = \frac{1}{u}\). Function notation helped us articulate that the operation \(\frac{1}{x+3}\) is actually a composition of these two simpler functions: \(f(g(x))\).
Clearly understanding function notation is crucial, as it ensures you can express and manipulate complex functions efficiently. This clarity makes it easier to verify results, as shown in the solution where substituting \(g(x)\) into \(f(u)\) verified that the composition gives us back the original \(Q(x)\).