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Chapter 2: Problem 54

Express the function in the form \(f \circ g \circ h\) $$G(x)=\frac{2}{(3+\sqrt{x})^{2}}$$

Short Answer

Expert verified
The function can be expressed as \( f \circ g \circ h \) with \( h(x) = \sqrt{x} \), \( g(x) = 3 + x \), \( f(x) = \frac{2}{x^2} \).

Step by step solution

01

Identify Inner Functions

First, observe the given function: \( G(x) = \frac{2}{(3 + \sqrt{x})^2} \). Notice that there are nested functions involving arithmetic operations and roots. We can express \( G(x) \) using three simpler functions. Start by identifying the innermost operation, which is the square root: \( h(x) = \sqrt{x} \).
02

Define the Second Layer Function

The second operation involves adding 3 to the result of the square root. Define this as the function \( g(x) = 3 + x \). Here, \( x \) represents the result from the previous function, which is \( h(x) \). So, \( g(h(x)) = 3 + \sqrt{x} \).
03

Define the Outer Layer Function

The outermost operation in the expression is taking the square of the result from the second function and using it as the denominator for 2. Define this function: \( f(x) = \frac{2}{x^2} \). This function takes the result of \( g(x) \) as input, giving us \( f(g(h(x))) = \frac{2}{(3 + \sqrt{x})^2} \).
04

Compose the Functions

Finally, compose these functions in the order of operations from innermost to outermost: \( f(g(h(x))) \). Substituting the earlier defined functions, we get \( f(g(h(x))) = \frac{2}{(3 + \sqrt{x})^2} \), confirming \( G(x) = f \circ g \circ h \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Inner Functions
Inner functions are fundamental in decomposing complex expressions. The concept focuses on identifying the innermost operations within a multi-step process. In our given exercise, the innermost task involves the operation of square roots. This step is crucial because by understanding the smallest unit of a problem, we can build up to more complicated solutions. In our example, we define:
  • Innermost function: Represented by taking the square root: \( h(x) = \sqrt{x} \)
Identifying and appropriately defining inner functions clarifies the structure of more sophisticated equations. Breaking down a function into these smaller parts allows easier manipulation and understanding, paving the way for defining outer functions that depend on these fundamental steps.
Nested Functions
Nested functions take the result of an inner function and use it as the input for another function. They are like building blocks, where the output of one becomes the input of the next. This setup helps streamline multifaceted operations into simpler, layered components. In our function, the second function takes the result of the square root and adds a constant:
  • Second function: Defined by adding 3 to the output of the square root: \( g(x) = 3 + x \), where \( x \) is the result of \( h(x) \)
This layering effect that nested functions create simplifies complex expressions by setting them out in a step-by-step fashion. Understanding this concept is crucial for handling processes where computations build upon one another incrementally.
Composing Functions
Composing functions involves creating a new function by applying one function to the result of another. It allows combining various operations into a single, elegant expression. For our problem, it involves bringing together the output from our inner and nested functions to compose the full function.
  • Composed function: The result is then taken from the output of the nested function and used in the final layer: \( f(x) = \frac{2}{x^2} \)
  • Here, \( f(g(h(x))) \) is the final composite expression, which excellently demonstrates the interplay between function layers: \( \frac{2}{(3 + \sqrt{x})^2} \)
Function composition enables tackling even sophisticated expressions and problems by breaking them into sequences of simpler functions. This approach reveals deeper mathematical structures that can simplify solving them, illustrating mathematics' beauty and logic.

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Most popular questions from this chapter

Find a function whose graph is the given curve. The bottom half of the circle \(x^{2}+y^{2}=9\)

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