Problem 17 - Calculus Single Variable

Question

Find a function \(g\) such that \(h=g \circ f\) \(h(x)=\left(x^{2}+1\right) /\left(x^{4}+2 x^{2}+3\right), f(x)=x^{2}+1\)

Step-by-Step Solution

Answer

The function \(g(u) = \frac{u}{u^2 + 2}\) satisfies \(h = g \circ f\).

Step 1: Understand the Composition

We are given that \(h = g \circ f\), which means \(h(x) = g(f(x))\). Here, \(h(x) = \frac{x^2 + 1}{x^4 + 2x^2 + 3}\) and \(f(x) = x^2 + 1\). We need to find a function \(g(u)\) such that \(h(x) = g(f(x))\).

Step 2: Simplify the Expression

Replace \(f(x)\) with \(u\), where \(u = x^2 + 1\). Now, express \(h(x)\) as \(\frac{f(x)}{x^4 + 2x^2 + 3}\).

Step 3: Recognize the Denominator Form

Note that the denominator, \(x^4 + 2x^2 + 3\), can be rewritten in terms of \(f(x)\) by substituting \(x^2 = u - 1\), resulting in \((u - 1)^2 + 2(u - 1) + 3\).

Step 4: Express Denominator in Terms of \(u\)

Simplify \((u - 1)^2 + 2(u - 1) + 3\) to get \((u^2 - 2u + 1) + 2u - 2 + 3 = u^2 + 2\).

Step 5: Rewrite \(h(x)\) in Terms of \(u\)

Given that \(f(x) = u\) and the denominator becomes \(u^2 + 2\), substitute back into the expression for \(h(x)\), which becomes \(\frac{u}{u^2 + 2}\).

Step 6: Determine \(g(u)\)

Now, equate \(g(u) = \frac{u}{u^2 + 2}\). This is the function \(g\) such that \(g(f(x)) = h(x)\), fulfilling the requirement that \(h = g \circ f\).

Key Concepts

Function Composition

Function composition is a fundamental concept in mathematics, particularly in function analysis. It combines two functions into a single function. When we say \( h = g \circ f \), it means the function \( h \) is the composition of \( g \) and \( f \). Simply put, you first apply function \( f \) to an input \( x \), and then the result of \( f(x) \) becomes the input for function \( g \). This chain-like process forms the composed function \( h(x) = g(f(x)) \).

In our exercise, understanding this means that the entire function \( h(x) = \frac{x^2 + 1}{x^4 + 2x^2 + 3} \) is created by applying the transformation \( f(x) = x^2 + 1 \) first, and then the resulting value is plugged into \( g \).
To grasp composition better, keep in mind the importance of order; the output from \( f \) is crucial as it directly dictates the input for \( g \). This mechanism can model diverse real-world scenarios, such as calculating distance traveled over time using velocity and time functions.

Algebraic Manipulation

Algebraic manipulation involves transforming mathematical expressions into more convenient forms. Here, it plays a crucial role in simplifying and understanding composed functions. Throughout the exercise, we utilized algebraic manipulation to express the denominator of \( h(x) \) entirely in terms of \( f(x) \).

Initially, we had the denominator \( x^4 + 2x^2 + 3 \), and through substitution, we managed to rewrite it as \( (u-1)^2 + 2(u-1) + 3 \) where \( u = x^2 + 1 \).
Upon expanding and simplifying, we obtained a cleaner expression: \( u^2 + 2 \).

This step of algebraic manipulation simplifies the entire function \( h(x) \) to \( \frac{u}{u^2 + 2} \), making it very straightforward to identify our desired function \( g(u) \).
Algebraic manipulation is essential for breaking down complex functions into smaller, more manageable pieces and finding relations between them.

Finding Functions

Identifying the function \( g \) that matches the criteria \( h = g \circ f \) completes the challenge of function composition. By focusing on rewriting \( h(x) \) in terms of \( f(x) \), we transitioned from \( h(x) = \frac{x^2 + 1}{x^4 + 2x^2 + 3} \) to a simpler form \( \frac{u}{u^2 + 2} \).

Recognizing this form is pivotal as it directly represents the function \( g(u) \). Thus, we equated \( g(u) = \frac{u}{u^2 + 2} \), fulfilling the condition where \( g(f(x)) = h(x) \).

• The process required recognizing patterns by substituting back into known transformations.
• Transformations involved isolating input functions and aligning them into the expected structure.

"Finding functions" often exercises critical skills such as pattern recognition and substitution. These skills can be applied to identifying unknown relationships between variables and solving complex mathematical models.