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Chapter 2: Problem 45

Express the function in the form \(f \circ g\) $$F(x)=(x-9)^{5}$$

Short Answer

Expert verified
Choose \(g(x) = x - 9\) and \(f(u) = u^5\) so that \(f(g(x)) = (x-9)^5\).

Step by step solution

01

Choose Inner Function

To express a function in the form \(f \circ g\), you need to identify an inner function \(g(x)\) that will be substituted into an outer function \(f(x)\). In this case, let's choose the inner function as \(g(x) = x - 9\).
02

Define Outer Function

With the inner function \(g(x) = x - 9\), we examine the remaining part of \(F(x) = (x-9)^5\). The outer function \(f(x)\) must take \(g(x)\) as its input and produce \(F(x)\). Thus, the outer function can be defined as \(f(u) = u^5\), where \(u = g(x)\).
03

Combine Functions

Now substitute the inner function \(g(x) = x - 9\) into the outer function \(f(u) = u^5\). This gives us \(f(g(x)) = f(x-9) = (x-9)^5\), which is equivalent to \(F(x)\).

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Key Concepts

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

Inner Function
The inner function is a key element when dealing with the composition of functions. It acts as a preliminary step within a larger function. Imagine picturing the inner function as a smaller gear turning within a larger machine. This smaller gear, or inner function, helps initiate the process. In our given exercise, the expression \((x-9)^5\) can be dissected to identify the inner component.
  • Here, the inner function is chosen as \(g(x) = x - 9\). It's like preparing the right ingredients before baking a cake. The inner function helps us see the part of the problem that will first influence how the entire function operates.
  • This function \(g(x) = x - 9\) simplifies the process by providing a base or foundation to work from before building onto it with more complex operations.
Understanding the inner function is about recognizing this essential starting point. Doing so allows us to effectively manage and set up the outer function that follows.
Outer Function
The outer function acts like the frame or structure that shapes the final result of a composition of functions. Imagine it as the overarching recipe that tells you what to do with your prepared ingredients, which is supplied by the inner function. Once you pin down the inner function, finding the outer function becomes a strategic move.
  • In our example, after defining \(g(x) = x - 9\) as the inner function, we look at the expression as a whole: \((x-9)^5\). To decipher what the outer function does, we notice it takes its input, the result of \(g(x)\), and raises it to the 5th power.
  • Thus, the outer function can be expressed as \(f(u) = u^5\). Here, \(u\) is the placeholder for the input from the inner function \(g(x)\).
The outer function effectively wraps itself around the final structure, determining how the inner function's output is transformed to produce the desired result.
Function Notation
Function notation is a shorthand representation in mathematics which makes solving equations involving functions more manageable. It uses letters like \(f\), \(g\), or \(h\), to easily reference functions without repeating lengthy expressions. This notation simplifies the process of combining functions and identifying their roles.
  • For our exercise, \(F(x)\) is expressed in the form \(f \circ g\), which means \(f(g(x))\). This highlights the sequence of operations 鈥� first applying \(g\) to \(x\) and then \(f\) to the output of \(g\).
  • Function notation helps clearly articulate each function's role in the composition, and enables precise manipulation and substitution within mathematical equations.
By using this notation format, we make it easier to understand and dissect complex functions, resulting in a clearer pathway to reach solutions. With practice, anyone can effectively navigate through functional compositions using this common mathematical language.

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Most popular questions from this chapter

Find \(f \circ g \circ h\) $$f(x)=\frac{1}{x}, \quad g(x)=x^{3}, \quad h(x)=x^{2}+2$$

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