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Chapter 1: Problem 45

Find a composite function form for \(y\) $$ y=\frac{1}{(x-3)^{4}} $$

Short Answer

Expert verified
The composite function is \( f(g(x)) = \frac{1}{(x-3)^4} \) with \( f(u) = \frac{1}{u^4} \) and \( g(x) = x - 3 \).

Step by step solution

01

Identify Inner and Outer Functions

To find the composite function form, identify two functions, say \( g(x) \) as the inner function and \( f(u) \) as the outer function. In this case, choose \( g(x) = x - 3 \) and \( f(u) = \frac{1}{u^4} \).
02

Substitute the Inner Function

Substitute \( g(x) \) into \( f(u) \) by setting \( u = g(x) = x - 3 \). This allows \( y = f(g(x)) = f(x - 3) = \frac{1}{(x - 3)^4} \).
03

Formulate the Composite Function

The composite function is \( (f \circ g)(x) \), which we've identified as \( f(g(x)) = \frac{1}{(x-3)^4} \). The functions \( f(u) = \frac{1}{u^4} \) and \( g(x) = x - 3 \) compose to form the original function.

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Key Concepts

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

Inner Function
To understand the idea of an inner function, think of it as the first step in a calculation that gets plugged into another function. In the given exercise, the inner function is represented by \( g(x) = x - 3 \). Here, the operation \( x - 3 \) simplifies an input \( x \) to a new value that will be utilized by the outer function.

Key points about inner functions:
  • The inner function is like the initial ingredient you need before proceeding to the next step.
  • It is the first function that operates on the input.
  • Identifying the inner function is crucial as it dictates what the outer function will act on.
In our scenario, applying the inner function \( g(x) \) means transforming any input \( x \) by subtracting 3.
Outer Function
The outer function is what takes the output from the inner function and further manipulates it to produce the final result. In the original exercise, the outer function is \( f(u) = \frac{1}{u^4} \). This means it is a function that performs a calculation on \( u \), which is itself the result of the inner function.

Ways to recognize and understand the outer function:
  • The outer function is applied last in the sequence of operations.
  • It uses the result generated by the inner function as its input.
  • The role of the outer function is to transform the intermediate result into the final output.
In practical terms, after applying \( g(x) = x - 3 \), the function \( f(u) = \frac{1}{u^4} \) processes \( u \) to provide \( \frac{1}{(x-3)^4} \). Each layer of operation is clearly defined by the outer function's structure.
Function Composition
Function composition is like building a computational path to solve a problem using two distinct functions consecutively. The procedure involves nesting one function inside another, leading to a composite function. In simpler terms, a composite function like \( (f \circ g)(x) \) implies that function \( g(x) \) is executed first, and its result is fed into function \( f(u) \).

Highlights of function composition:
  • It combines two functions where the output of one becomes the input for the other.
  • The notation \( (f \circ g)(x) \) reads as "\( f \) of \( g \) of \( x \)".
  • Evaluating a composite function involves substituting the inner function into the outer one.
For the given problem, the composite function is \( (f \circ g)(x) = \frac{1}{(x-3)^4} \). By understanding this, you learn how separate functions can be effectively used together to model complex relationships.

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