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Chapter 2: Problem 30

Find the rules for the composite functions \(f \circ g\) and \(g \circ f\). \(f(x)=\sqrt{x+1} ; g(x)=\frac{1}{x-1}\)

Short Answer

Expert verified
The rules for the composite functions are: \(f \circ g(x) = \sqrt{\frac{x}{x-1}}\) \(g \circ f(x) = \frac{1}{\sqrt{x+1} - 1}\)

Step by step solution

01

Find the rule for \(f \circ g(x)\)

To find the rule for \(f \circ g(x)\), substitute the output of function \(g\) into function \(f\): \(f(g(x)) = f\left(\frac{1}{x-1}\right)\) Now, replace \(x\) in the function \(f(x) = \sqrt{x+1}\) with the output of function \(g\), which is \(\frac{1}{x-1}\): \(f(g(x)) = \sqrt{\frac{1}{x-1}+1}\)
02

Simplify the expression for \(f(g(x))\)

To simplify the expression for \(f(g(x))\), combine the fractions under the square root \(f(g(x)) = \sqrt{\frac{1}{x-1}+\frac{x-1}{x-1}}\) Now combine the fractions under the square root: \(f(g(x)) = \sqrt{\frac{1+(x-1)}{(x-1)}}\) Simplify the numerator: \(f(g(x)) = \sqrt{\frac{x}{x-1}}\) The rule for the composite function of \(f \circ g(x)\) is: \(f \circ g(x) = \sqrt{\frac{x}{x-1}}\)
03

Find the rule for \(g \circ f(x)\)

To find the rule for \(g \circ f(x)\), substitute the output of function \(f\) into function \(g\): \(g(f(x)) = g(\sqrt{x+1})\) Now, replace \(x\) in the function \(g(x) = \frac{1}{x-1}\) with the output of function \(f\), which is \(\sqrt{x+1}\): \(g(f(x)) = \frac{1}{\sqrt{x+1} - 1}\) The rule for the composite function of \(g \circ f(x)\) is: \(g \circ f(x) = \frac{1}{\sqrt{x+1} - 1}\)
04

State the rules of the composite functions \(f \circ g(x)\) and \(g \circ f(x)\)

The rules for the composite functions are: \(f \circ g(x) = \sqrt{\frac{x}{x-1}}\) \(g \circ f(x) = \frac{1}{\sqrt{x+1} - 1}\)

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Key Concepts

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

Function Composition
Function composition refers to the process of applying one function to the results of another. Think of it like putting together building blocks, where the output of one function becomes the input for the next one. This order of operations is key! If we have two functions, say \(f(x)\) and \(g(x)\), the composition \(f \circ g\) means you first apply \(g\) and then \(f\). For our specific problem, to find \(f \circ g(x)\), substitute \(g(x)\) into \(f(x)\), leading to the expression \(f(g(x)) = \sqrt{\frac{1}{x-1} + 1}\). Here, \(g\) alters the original input before \(f\) processes it.
Function composition allows us to create new functions with unique expressions and properties.
  • Not everyone notices at first, but the order you compose matters a lot. \(f\circ g\) usually isn't the same as \(g\circ f\).
  • Composed functions can simplify or solve complex problems by "chaining" operations.
Function Operations
When dealing with functions, function operations refer to the various ways in which functions can interact with each other. Besides composition, other basic operations include addition, subtraction, multiplication, and division of functions. These operations combine two functions to create a new one, allowing for a myriad of possibilities.
Let's explore their distinctions briefly:
  • Addition and Subtraction: Functions can be added or subtracted by simply combining their outputs: \((f + g)(x) = f(x) + g(x)\), \((f - g)(x) = f(x) - g(x)\).
  • Multiplication and Division: Similarly, functions multiply or divide: \((f \cdot g)(x) = f(x) \cdot g(x)\), \((f / g)(x) = \frac{f(x)}{g(x)}\), provided \(g(x)\) isn't zero.
  • Composition: In contrast to basic arithmetic operations, composition means inserting the output of one function into another, requiring a deeper understanding of the function's domain and range.
Function operations expand our toolkit for manipulating and transforming functions, allowing for more mathematical creativity and problem-solving versatility.
Mathematical Functions
Mathematical functions are the foundation of many mathematical operations and models. A function is a special relationship where each input has a single output. Think of it like a vending machine, where each button you press dispenses exactly one specific snack. In math, this is about pairing an input with one definitive output.
Functions come in different forms, such as linear, quadratic, exponential, and more. Each has its own distinctive formulae and properties.
  • The power of functions lies in their ability to model real-world situations and abstract mathematical concepts.
  • With functions like \(f(x) = \sqrt{x+1}\) and \(g(x) = \frac{1}{x-1}\), we explore the relationships between variables and craft new mathematical expressions by composing them.
Functions form the heart of mathematical analysis and computation, serving as the building blocks for equations and models throughout math and science.

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Most popular questions from this chapter

DECISION ANALYSIS A product may be made using machine I or machine II. The manufacturer estimates that the monthly fixed costs of using machine I are \(\$ 18,000\), whereas the monthly fixed costs of using machine II are \(\$ 15,000\). The variable costs of manufacturing 1 unit of the product using machine I and machine II are \(\$ 15\) and \(\$ 20\), respectively. The product sells for \(\$ 50\) each. a. Find the cost functions associated with using each machine. b. Sketch the graphs of the cost functions of part (a) and the revenue functions on the same set of axes. c. Which machine should management choose in order to maximize their profit if the projected sales are 450 units? 550 units? 650 units? d. What is the profit for each case in part (c)?

AvERAGE SINGLE-FAMILY PROPERTY TAX Based on data from 298 of 351 cities and towns in Massachusetts, the average single-family tax bill from 1997 through 2007 is approximated by the function $$ T(t)=7.26 t^{2}+91.7 t+2360 \quad(0 \leq t \leq 10) $$ where \(T(t)\) is measured in dollars and \(t\) in years, with \(t=0\) corresponding to 1997 . a. What was the property tax on a single-family home in Massachusetts in \(1997 ?\) b. If the trend continues, what will be the property tax in \(2010 ?\)

According to a study conducted in 2004, the number of subscribers of BlackBerry, the handheld email devices manufactured by Research in Motion Ltd., is approximated by \(N(t)=-0.0675 t^{4}+0.5083 t^{3}-0.893 t^{2}+0.66 t+0.32\) \((0 \leq t \leq 4)\) where \(N(t)\) is measured in millions and \(t\) in years, with \(t=0\) corresponding to the beginning of 2002 . a. How many BlackBerry subscribers were there at the beginning of \(2002 ?\) b. How many BlackBerry subscribers were there at the beginning of \(2006 ?\)

As broadband Internet grows more popular, video services such as YouTube will continue to expand. The number of online video viewers (in millions) is projected to grow according to the rule $$ N(t)=52 t^{0.531} \quad(1 \leq t \leq 10) $$ where \(t=1\) corresponds to the beginning of 2003 . a. Sketch the graph of \(N\). b. How many online video viewers will there be at the beginning of \(2010 ?\)

In Exercises 19-24, find the points of intersection of the graphs of the functions. \(f(x)=-x^{2}+4 ; g(x)=x+2\)
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