Problem 95
Question
In Exercises \(87-96,\) find two functions \(f\) and \(g\) such that \(h(x)=(f \circ g)(x)=f(g(x)) .\) Answers may vary. $$h(x)=4(2 x+9)^{5}-(2 x+9)^{8}$$
Step-by-Step Solution
Verified Answer
The two functions that satisfy the composition condition \(h(x) = (f \circ g)(x) = f(g(x))\) for the given \(h(x) = 4(2x+9)^5 - (2x+9)^8\) are \(g(x) = 2x + 9\) and \(f(x) = 4x^5 - x^8\).
1Step 1: Determine the inner function
The inner function \(g(x)\) can usually be spotted as the function that is used inside of another function in the equation. Here, the occurrence of \(2x + 9\) inside the exponential values reveals the function \(g(x) = 2x + 9\).
2Step 2: Determine the outer function
The outer function \(f(x)\) is the main function that operates on the result of the inner function, \(g(x)\). Given the chosen inner function, we can rewrite the equation as \(f(g(x)) = 4(g(x))^5 - (g(x))^8\). This leads to the identification of the outer function as \(f(x) = 4x^5 - x^8\).
3Step 3: Verify composition of functions
To assure the correctness of the chosen functions, substitute \(g(x) = 2x + 9\) into the outer function \(f(x)\) in the form of \(f(g(x))\) which gives \(f(g(x)) = 4(2x+9)^5-(2x+9)^8\). This expression is the same as the original equation for \(h(x)\), hence the choice of functions is correct.
Key Concepts
Inner FunctionOuter FunctionFunction NotationVerification of Functions
Inner Function
In function composition, the inner function is the first function that operates on the input. It is typically found within another function in an expression. For example, in the expression \(h(x) = f(g(x))\), the \(g(x)\) part represents the inner function. It is where we begin when constructing a function composition.
To identify the inner function, look for consistent expressions or terms repeated in the operation. In our exercise, the expression \(2x + 9\) appears prominently inside the higher power terms. This makes it a suitable choice for \(g(x)\), defined as \(g(x) = 2x + 9\). By recognizing patterns, we simplify our understanding and gain clarity on the function composition.
Understanding the inner function's role helps in breaking down complex equations, as it provides the initial modification to the input before any other processes take place.
To identify the inner function, look for consistent expressions or terms repeated in the operation. In our exercise, the expression \(2x + 9\) appears prominently inside the higher power terms. This makes it a suitable choice for \(g(x)\), defined as \(g(x) = 2x + 9\). By recognizing patterns, we simplify our understanding and gain clarity on the function composition.
Understanding the inner function's role helps in breaking down complex equations, as it provides the initial modification to the input before any other processes take place.
Outer Function
The outer function in a composition is a function that takes the result from the inner function as its input. It applies further transformations or operations on this output. In terms of notation, if \(g(x)\) is the inner function, then \(f(x)\) becomes the outer function in \(f(g(x))\).
Consider our given composition \(h(x) = f(g(x)) = 4(g(x))^5 - (g(x))^8\). We found that \(g(x) = 2x + 9\). To determine the outer function, substitute this back into the expression:
\(f(g(x)) = 4(2x+9)^5 - (2x+9)^8\).
Everything operating on \(g(x)\) is part of \(f(x)\), leading to \(f(x) = 4x^5 - x^8\). This function tells us how the result from the inner function should be altered next.
Consider our given composition \(h(x) = f(g(x)) = 4(g(x))^5 - (g(x))^8\). We found that \(g(x) = 2x + 9\). To determine the outer function, substitute this back into the expression:
\(f(g(x)) = 4(2x+9)^5 - (2x+9)^8\).
Everything operating on \(g(x)\) is part of \(f(x)\), leading to \(f(x) = 4x^5 - x^8\). This function tells us how the result from the inner function should be altered next.
Function Notation
Function notation is quite versatile, allowing us to represent complex operations simply by using symbols like \(f(x)\). This notation essentially communicates how we transform the input (often represented as \(x\)) into an output through a defined rule.
The notation becomes even more powerful when dealing with compositions. For example, in \(h(x) = f(g(x))\), each "parenthesis" signifies an operation. This allows us to express the transformation process in a clear, step-by-step manner: \(g(x)\), then \(f(g(x))\), all while maintaining any modifications we make to the initial input.
Function notation is crucial when solving problems in mathematics, as it simplifies the way we handle and write down these transformations.
The notation becomes even more powerful when dealing with compositions. For example, in \(h(x) = f(g(x))\), each "parenthesis" signifies an operation. This allows us to express the transformation process in a clear, step-by-step manner: \(g(x)\), then \(f(g(x))\), all while maintaining any modifications we make to the initial input.
Function notation is crucial when solving problems in mathematics, as it simplifies the way we handle and write down these transformations.
Verification of Functions
Verification is an essential step in mathematics to confirm that the functions we've derived are correctly composed to form the target expression. To verify, we replace \(g(x)\) into the outer function equation and simplify to check our original \(h(x)\) identity.
For this exercise, we check whether \(f(g(x))\) truly reproduces the original expression, \(h(x) = 4(2x+9)^5 - (2x+9)^8\). Replace the known \(g(x) = 2x + 9\) into \(f(x) = 4x^5 - x^8\), yielding:
\(f(g(x)) = 4(2x+9)^5 - (2x+9)^8\).
The simplification shows that \(f(g(x))\) matches the provided \(h(x)\), confirming our identified inner and outer functions are indeed accurate. This process reassures that no steps have been overlooked and that our function breakdown is logically sound.
For this exercise, we check whether \(f(g(x))\) truly reproduces the original expression, \(h(x) = 4(2x+9)^5 - (2x+9)^8\). Replace the known \(g(x) = 2x + 9\) into \(f(x) = 4x^5 - x^8\), yielding:
\(f(g(x)) = 4(2x+9)^5 - (2x+9)^8\).
The simplification shows that \(f(g(x))\) matches the provided \(h(x)\), confirming our identified inner and outer functions are indeed accurate. This process reassures that no steps have been overlooked and that our function breakdown is logically sound.
Other exercises in this chapter
Problem 94
Is it possible for a quadratic function with real coefficients to have one real zero and one nonreal zero? Explain. (Hint: Examine the quadratic formula.)
View solution Problem 95
A parabola associated with a certain quadratic function \(f\) has the point (2,8) as its vertex and passes through the point \((4,0) .\) Find an expression for
View solution Problem 96
Is it possible for a quadratic function to have the set of all real numbers as its range? Explain. (Hint: Examine the graph of a general quadratic function.)
View solution Problem 96
In Exercises \(87-96,\) find two functions \(f\) and \(g\) such that \(h(x)=(f \circ g)(x)=f(g(x)) .\) Answers may vary. $$h(x)=(3 x-7)^{10}+5(3 x-7)^{2}$$
View solution