/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 41 Find (a) \(f \circ g\) and (b) \... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 41

Find (a) \(f \circ g\) and (b) \(g \circ f .\) Find the domain of each function and each composite function. $$f(x)=\frac{1}{x}, \quad g(x)=x+3$$

Short Answer

Expert verified
\((f \circ g)(x) = 1/(x+3)\) with domain \((- \infty, -3) U (-3, +\infty)\), and \((g \circ f)(x) = (1/x) + 3\) with domain \((- \infty, 0) U (0, +\infty)\)

Step by step solution

01

Compute Composite Functions

To compute \(f \circ g\), substitute \(x+3\) (which is \(g(x)\)) into \(f(x)\). This gives \((f \circ g)(x) = f(g(x)) = 1/(x+3)\). To compute \(g \circ f\), substitute \(1/x\) (which is \(f(x)\)) into \(g(x)\). This gives \((g \circ f)(x) = g(f(x)) = (1/x) + 3\).
02

Find Domains of the Functions

For \(f(x)\), the domain can't include 0 as it results in division by zero. So, \(Dom(f) = (-\infty, 0) 鈭� (0, +\infty)\). For \(g(x)\), the domain is the set of all real numbers since there isn't any restriction to what \(x\) can be. Therefore, \(Dom(g) = (-\infty, +\infty)\).
03

Find Domains of the Composite Functions

For \(f \circ g = 1/(x+3)\), \(x\) can't be -3 because it would involve division by zero. Therefore, its domain is \(Dom(f \circ g) = (-\infty, -3) 鈭� (-3, +\infty)\). For \(g \circ f = (1/x) + 3\), \(x\) can't be 0 because it would involve division by zero. So, its domain is \(Dom(g \circ f) = (-\infty, 0) 鈭� (0, +\infty)\).
04

Final Answer

(a) \(f \circ g = 1/(x+3)\) with domain \((- \infty, -3) U (-3, +\infty)\) and (b) \(g \circ f = (1/x) + 3\) with domain \((- \infty, 0) U (0, +\infty)\).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Find the difference quotient and simplify your Answer: $$f(t)=\frac{1}{t-2}, \quad \frac{f(t)-f(1)}{t-1}, \quad t \neq 1$$

Use a graphing utility to graph the function. Be sure to choose an appropriate viewing window. $$f(x)=2.5 x-4.25$$

Determine whether the statement is true or false. Justify your answer. The set of ordered pairs \(\\{(-8,-2),(-6,0),(-4,0)\) \((-2,2),(0,4),(2,-2)\\}\) represents a function.

Sketch the graph of the function. $$g(x)=-[[x]]$$

For groups of 80 or more people, a charter bus company determines the rate per person according to the formula Rate \(=8-0.05(n-80), \quad n \geq 80\) where the rate is given in dollars and \(n\) is the number of people. (a) Write the revenue \(R\) for the bus company as a function of \(n\) (b) Use the function in part (a) to complete the table. What can you conclude? $$\begin{array}{|l|l|l|l|l|l|l|l|}\hline n & 90 & 100 & 110 & 120 & 130 & 140 & 150 \\\\\hline R(n) & & & & & & & \\\\\hline\end{array}$$
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Decision Maths

Read Explanation

Logic and Functions

Read Explanation

Geometry

Read Explanation

Discrete Mathematics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.