Problem 51 - Algebra and Trigonometry

Question

Express the function in the form \(f \circ g\) $$ G(x)=\frac{x^{2}}{x^{2}+4} $$

Step-by-Step Solution

Answer

\( f(u) = \frac{u}{u+4} \) and \( g(x) = x^2 \).

Step 1: Understanding the Function

We start by analyzing the function \( G(x) = \frac{x^{2}}{x^{2}+4} \). We need to express it as a composition of two functions, \( f \) and \( g \), such that \( G(x) = f(g(x)) \).

Step 2: Defining the Composite Function's Inner Function

Let's define the inner function \( g(x) \). Notice that \( g(x) = x^2 \) is inside both the numerator and the denominator of the given function. Therefore, let \( g(x) = x^2 \). This simplifies our task since \( g(x) \) represents the core transformation applied to \( x \).

Step 3: Defining the Composite Function's Outer Function

Once we have \( g(x) = x^2 \), we must define \( f(u) \) using \( u = g(x) = x^2 \). From \( G(x) = \frac{x^{2}}{x^{2}+4} \), we see that \( f(u) = \frac{u}{u+4} \). This is because substituting \( g(x) = x^2 \) into \( f(u) \) gives us back the original function.

Step 4: Verification of the Composition

We verify that \( G(x) = f(g(x)) \). Substitute \( g(x) = x^2 \) into \( f(u) = \frac{u}{u+4} \) to get \( f(g(x)) = \frac{x^2}{x^2+4} \). This matches the given function \( G(x) \), confirming that the composition is correct.

Key Concepts

Inner Function

When working with function composition, understanding the concept of an inner function is essential. In a composite function, the inner function is the first one you apply to the input variable. It is known as the 'core transformation' because it takes the input and processes it before passing it on to the next function.

In the exercise, we express the function \( G(x) = \frac{x^{2}}{x^{2}+4} \) as a composition of two functions, \( f \) and \( g \). The inner function, \( g(x) \), is determined by observing what transformation is common between both the numerator and the denominator. Here, it’s \( g(x) = x^2 \), representing the transformation applied to \( x \).

• Transforms the input first
• Completed before the outer function
• Determines the starting point of the composition

Understanding the inner function simplifies solving complex compositions, as it forms the foundation of the overall function structure.

Outer Function

Once you have figured out the inner function, the outer function becomes the next puzzle piece in function composition. The outer function takes the result of the inner function as its input.

In our example, once we have identified the inner function as \( g(x) = x^2 \), we then find the outer function \( f(u) \) by using \( u = g(x) \, or \, u = x^2 \). For the given function \( G(x) \), we can see that inserting \( g(x) = x^2 \) into the function results in \( f(u) = \frac{u}{u+4} \).

• Receives the processed input from the inner function
• Completes the function composition
• Represents the overall behavior of the composite function

By understanding the outer function, you can grasp the aggregate impact on the original variable after both transformations have taken place.

Composite Function

The composite function is a beautiful culmination of both the inner and the outer functions.

Composing functions means you apply one function to the results of another, creating a seamless operation. In this finalized picture, you see how each function plays a vital role, combining their effects to create a new function. The exercise involves expressing \( G(x) = \frac{x^{2}}{x^{2}+4} \) as \( f(g(x)) \), showing that this technique simplifies complex calculations through a step-by-step breakdown.

• Represents the complete function pairing
• Simplifies calculations by clarifying component functions
• Facilitates advanced mathematical understanding

Seeing how individual transformations combine to address holistic change is the essence of appreciating composite functions. Once mastered, it enriches your problem-solving skills in calculus and beyond.