Problem 31
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.) $$p(x)=(2 x+3)^{3}$$
Step-by-Step Solution
Verified Answer
Write \(p(x)\) as \(p(x) = f(g(x))\) where \(f(x) = x^3\) and \(g(x) = 2x + 3\).
1Step 1: Identify the Inner Function
To find a composition of functions, start by identifying the inside function of the given expression. In \(p(x)=(2x+3)^3\), the inner part is \(2x+3\). Let's call this function \(g(x)\). Therefore, \(g(x) = 2x + 3\).
2Step 2: Define the Outer Function
Now, define the outer function based on the form of \(p(x)\). After substituting \(g(x)\) into \(p(x)\), the expression becomes \((g(x))^3\). So, define the outer function \(f(x)\) such that \(f(x) = x^3\).
3Step 3: Compose the Functions
Write the given function \(p(x)\) as a composition of \(f(x)\) and \(g(x)\). Therefore, \(p(x)\) can be expressed as \((f \circ g)(x) = f(g(x)) = (2x+3)^3\). This confirms that \(f(x) = x^3\) and \(g(x) = 2x + 3\).
Key Concepts
Inner FunctionOuter FunctionComposing FunctionsNon-Identity Functions
Inner Function
When discussing function composition, the term "inner function" is a pivotal part of the process. It refers to the function that is nested inside another function. Breaking it down simplifies the given expression. By identifying the inner function, you get closer to understanding how the overall function operates.
In the example of the function \(p(x) = (2x + 3)^3\), the inner part is the expression inside the parentheses,\(2x + 3\). This is the part that gets processed first, before applying any other operations.
The inner function sets the stage for applying the outer function. It essentially provides input to the next layer of computation. Understand the inner function well, as it tells how the input variable is initially modified.
In the example of the function \(p(x) = (2x + 3)^3\), the inner part is the expression inside the parentheses,\(2x + 3\). This is the part that gets processed first, before applying any other operations.
- The inner function is often expressed as \(g(x)\).
- In this case, \(g(x) = 2x + 3\).
The inner function sets the stage for applying the outer function. It essentially provides input to the next layer of computation. Understand the inner function well, as it tells how the input variable is initially modified.
Outer Function
Once the inner function is recognized, identifying the "outer function" is the next step. This function operates on the result of the inner function, further transforming it to derive the final output. In our expression \(p(x) = (2x + 3)^3\), after solving for the inner function, we look at how the entire expression is processed.
The outer function in this case works on the result \((g(x))\) obtained earlier:
This function takes the result of the inner function and raises it to the third power. Recognizing the outer function is crucial because it tells us what is ultimately done with the original expression's result. This layering concept is foundational in understanding more complex mathematical operations.
The outer function in this case works on the result \((g(x))\) obtained earlier:
- The outer function is denoted as \(f(x)\).
- Here, \(f(x) = x^3\).
This function takes the result of the inner function and raises it to the third power. Recognizing the outer function is crucial because it tells us what is ultimately done with the original expression's result. This layering concept is foundational in understanding more complex mathematical operations.
Composing Functions
Composing functions is the process of combining two or more functions to form a new function. It involves using the output of one function as the input to another. This is a common concept in mathematics, helping in creating complex expressions from simpler parts.
In our example, when we compose the functions, we write:
\[(f \circ g)(x) = f(g(x))\]
Using the expressions we've identified:
\[(f \circ g)(x) = (2x + 3)^3\]
This means you first apply \(g(x)\) to \(x\) and then apply \(f(x)\) to the result. Composing functions is a methodical way of building new functions and is useful in calculus, programming, and other fields where operations are layered.
In our example, when we compose the functions, we write:
\[(f \circ g)(x) = f(g(x))\]
Using the expressions we've identified:
\[(f \circ g)(x) = (2x + 3)^3\]
This means you first apply \(g(x)\) to \(x\) and then apply \(f(x)\) to the result. Composing functions is a methodical way of building new functions and is useful in calculus, programming, and other fields where operations are layered.
Non-Identity Functions
A non-identity function is a function that does more than merely output the input unchanged. These functions modify or transform the input in some way. In composition, non-identity functions are crucial because they allow for pattern and transformation analysis.
In the expression given, both \(g(x) = 2x + 3\) and \(f(x) = x^3\) are non-identity functions because:
Non-identity functions are integral to creating meaningful expressions. They help make function composition a powerful tool, allowing the creation of varied results.
In the expression given, both \(g(x) = 2x + 3\) and \(f(x) = x^3\) are non-identity functions because:
- \(g(x)\) transforms \(x\) by multiplying by 2 and adding 3.
- \(f(x)\) transforms the input by cubing it.
Non-identity functions are integral to creating meaningful expressions. They help make function composition a powerful tool, allowing the creation of varied results.
Other exercises in this chapter
Problem 31
Solve the equation or inequality. $$2 x^{-\frac{1}{3}}(x-3)^{\frac{1}{3}}+x^{\frac{2}{3}}(x-3)^{-\frac{2}{3}} \geq 0$$
View solution Problem 31
Let \(f\) and \(g\) be invertible functions. With the help of your classmates show that \((f \circ g)\) is one-to-one, hence invertible, and that \((f \circ g)^
View solution Problem 32
Solve the equation or inequality. $$\sqrt[3]{x^{3}+3 x^{2}-6 x-8}>x+1$$
View solution Problem 32
What graphical feature must a function \(f\) possess for it to be its own inverse?
View solution