Problem 47
Question
\(45-50\) Express the function in the form \(f \circ g\) $$ G(x)=\frac{x^{2}}{x^{2}+4} $$
Step-by-Step Solution
Verified Answer
\( G(x) = f(g(x)) \) with \( g(x) = x^2 \) and \( f(u) = \frac{u}{u+4} \).
1Step 1: Identify Inner Function
To write the function in the form of \( f(g(x)) \), start by identifying a potential inner function, \( g(x) \), that will simplify the original function. In this case, let \( g(x) = x^2 \).
2Step 2: Define the Outer Function
With \( g(x) = x^2 \) determined as the inner function, express \( G(x) \) in terms of \( g(x) \). Substituting \( g(x) \) into the original function gives \( G(x) = \frac{g(x)}{g(x) + 4} \). This means the outer function \( f(u) \) where \( u = g(x) \) is \( f(u) = \frac{u}{u + 4} \).
3Step 3: Verify the Decomposition
Check that substituting \( g(x) = x^2 \) into \( f(g(x)) \) results in the original function \( G(x) \). Substitute \( x^2 \) for \( u \) in \( f(u) \), which results in \( \frac{x^2}{x^2 + 4} \), confirming the decomposition is correct.
Key Concepts
Inner FunctionOuter FunctionFunction DecompositionMathematics Education
Inner Function
In the context of function composition, the inner function is an essential building block. It is the initial function applied to the input value. To simplify the process of expressing complex functions, identifying the inner function is a crucial first step.
This function, often noted as \( g(x) \), takes the input \( x \) and processes it to form an intermediate value. From the original exercise, \( g(x) = x^2 \) acts as our inner function. Defining this clearly helps to structure how the overall composition takes shape and allows us to manage calculations efficiently.
This function, often noted as \( g(x) \), takes the input \( x \) and processes it to form an intermediate value. From the original exercise, \( g(x) = x^2 \) acts as our inner function. Defining this clearly helps to structure how the overall composition takes shape and allows us to manage calculations efficiently.
- Begins the function evaluation process.
- Transforms the input value for further operations.
- Serves as the input for the outer function.
Outer Function
The outer function operates on the result of the inner function. Once the input has been transformed by the inner function, the outer function then uses this modified value to produce the final outcome. This is typically specified as \( f(u) \) in function composition.
In our example, after passing through the inner function \( g(x) = x^2 \), the value becomes \( u = x^2 \). The outer function \( f(u) \) then takes over, defined in the exercise as \( f(u) = \frac{u}{u + 4} \). The outer function integrates the intermediate result into a larger operation, effectively dictating the final result of the composition.
In our example, after passing through the inner function \( g(x) = x^2 \), the value becomes \( u = x^2 \). The outer function \( f(u) \) then takes over, defined in the exercise as \( f(u) = \frac{u}{u + 4} \). The outer function integrates the intermediate result into a larger operation, effectively dictating the final result of the composition.
- Final step in producing the outcome.
- Processes the result of the inner function.
- Essential in defining the shape and behavior of the composed function.
Function Decomposition
Function decomposition is the reverse process of function composition and involves breaking down a complex function into simpler, constituent parts. This technique is vital for understanding, analyzing, and solving mathematical problems efficiently.
In the solution provided, the original function \( G(x) = \frac{x^2}{x^2 + 4} \) is broken into two parts: \( g(x) = x^2 \) and \( f(u) = \frac{u}{u + 4} \). This makes handling and interpreting the function much easier, as each part can be studied and simplified independently.
In the solution provided, the original function \( G(x) = \frac{x^2}{x^2 + 4} \) is broken into two parts: \( g(x) = x^2 \) and \( f(u) = \frac{u}{u + 4} \). This makes handling and interpreting the function much easier, as each part can be studied and simplified independently.
- Helps simplify complex functions for easier manipulation.
- Facilitates checking and verifying of calculations.
- Useful in problem-solving and conceptual understanding.
Mathematics Education
Teaching the concepts of function composition and decomposition is pivotal in mathematics education. These ideas form the core of more complex mathematical topics, allowing students to advance their understanding of mathematics efficiently.
When educators introduce functions, they often start with simple operations, then build upon those foundations by exploring composed functions. By breaking down each function using inner and outer functions, students grasp more intuitively the mechanical and theoretical aspects of mathematics.
Practically, function composition demonstrates:
When educators introduce functions, they often start with simple operations, then build upon those foundations by exploring composed functions. By breaking down each function using inner and outer functions, students grasp more intuitively the mechanical and theoretical aspects of mathematics.
Practically, function composition demonstrates:
- How smaller functions interact to form a complete operation.
- Strategic thinking and problem-solving skills.
- Deeper insights into mathematical structures and their applications.
Other exercises in this chapter
Problem 46
Find the inverse function of \(f\). \(f(x)=\left(2-x^{3}\right)^{5}\)
View solution Problem 46
Find the domain of the function. $$ f(x)=\sqrt[4]{x+9} $$
View solution Problem 47
Sketch the graph of the piecewise defined function. $$ f(x)=\left\\{\begin{array}{ll}{0} & {\text { if }|x| \leq 2} \\ {3} & {\text { if }|x|>2}\end{array}\righ
View solution Problem 47
33–48 ? Sketch the graph of the function, not by plotting points, but by starting with the graph of a standard function and applying transformations. $$ y=|x+2|
View solution