Problem 35 - Precalculus

Question

Write the given function as a composition of two or more non-identity functions. (There are several correct answers, so check your answer using function composition.) $$r(x)=\frac{2}{5 x+1}$$

Step-by-Step Solution

Answer

The function can be composed as \( r(x) = r(g(x)) \) where \( r(u) = \frac{2}{u} \) and \( g(x) = 5x + 1 \).

Step 1: Identify the Outer Function

We begin by determining one function that can serve as the outer part of the composition. For the given function \( r(x) = \frac{2}{5x+1} \), notice that \( r(u) = \frac{2}{u} \) can be an outer function, where \( u \) is an intermediate expression.

Step 2: Identify the Inner Function

Having established that the outer function can be \( r(u) = \frac{2}{u} \), we observe that the expression \( 5x + 1 \) can act as the inner function. Thus, let \( g(x) = 5x + 1 \).

Step 3: Verify the Composition

To verify the composition, substitute the inner function into the outer function: \( r(g(x)) = r(5x+1) = \frac{2}{5x+1} \). This confirms that the original function \( r(x) \) can indeed be expressed as a composition of the two functions \( r(u) = \frac{2}{u} \) and \( g(x) = 5x + 1 \).

Key Concepts

Outer Function

In the realm of function composition, the outer function is an integral component of dissecting complex functions into simpler parts. When we view a function as a composition of two functions, the outer function typically operates on the result of the inner function. In our given exercise, we start by examining the function \( r(x) = \frac{2}{5x+1} \). Here, we identify the outer function by considering how the entire expression can be simplified.
For our specific function, \( r(x) \), one way to view its construction is through \( r(u) = \frac{2}{u} \), where \( u \) acts as a placeholder for an inner expression. This outer function, \( r(u) \), is a simpler version that reveals the division that dominates the entire original expression.
In simpler terms:

• The outer function determines the broad "shape" or final result form of your original function.
• It is like the framework or skeleton around which the original complex function wraps itself.

Inner Function

To understand the construction of a complex function, recognizing the inner function is equally crucial. An inner function is essentially the building block upon which the outer function acts.
In our example, after identifying the outer function \( r(u) = \frac{2}{u} \), the next step is finding the inner function. This is the expression that will replace \( u \) in the outer function, resulting in our original function.
For \( r(x) = \frac{2}{5x+1} \), the expression \( 5x + 1 \) naturally presents itself as this inner component. We express it as \( g(x) = 5x + 1 \).
Key insights into the inner function include:

• It acts as the 'ingredient' that gets processed by the outer function.
• Its result becomes the input to the outer function when computing the composed function.

Verify Composition

With both the outer and inner functions established, verifying their composition reassures us of the correct breakdown of the original function. Verification ensures that by substituting the inner function into the outer function, we arrive back at the starting function.
This verification procedure involves substituting \( g(x) = 5x + 1 \) into the outer function \( r(u) = \frac{2}{u} \). Substituting gives us \( r(g(x)) = r(5x + 1) = \frac{2}{5x + 1} \).
As seen, upon substituting, the result matches our initial function \( r(x) = \frac{2}{5x + 1} \), confirming that our composition is accurate.
Important points about verifying composition include:

• It validates the breakdown and reconstruction of a function ensuring correctness.
• Reaffirming the composition particularly helps in avoiding errors in identifying the function parts.