Problem

Explain, using Theorems 4, 5, 7, and 9, why the f…

05:22

Question

Answered step-by-step

Problem 26 Easy Difficulty

Explain, using Theorems 4, 5, 7, and 9, why the function is continuous at every number in its domain. State the domain.

$ G(x) = \dfrac{x^2 + 1}{2x^2 - x - 1} $

Video Answer

Solved by verified expert

Textbook Answer

Official textbook answer

Video by Linda Hand

Numerade Educator

This textbook answer is only visible when subscribed! Please subscribe to view the answer

Watch More Solved Questions in Chapter 2

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

Problem 6

Problem 7

Problem 8

Problem 9

Problem 10

Problem 11

Problem 12

Problem 13

Problem 14

Problem 15

Problem 16

Problem 17

Problem 18

Problem 19

Problem 20

Problem 21

Problem 22

Problem 23

Problem 24

Problem 25

Problem 26

Problem 27

Problem 28

Problem 29

Problem 30

Problem 31

Problem 32

Problem 33

Problem 34

Problem 35

Problem 36

Problem 37

Problem 38

Problem 39

Problem 40

Problem 41

Problem 42

Problem 43

Problem 44

Problem 45

Problem 46

Problem 47

Problem 48

Problem 49

Problem 50

Problem 51

Problem 52

Problem 53

Problem 54

Problem 55

Problem 56

Problem 57

Problem 58

Problem 59

Problem 60

Problem 61

Problem 62

Problem 63

Problem 64

Problem 65

Problem 66

Problem 67

Problem 68

Problem 69

Problem 70

Problem 71

Problem 72

Problem 73

Video Transcript

Okay we're given a function. And the the initial question is um why is it continuous at every point in its domain? And then it has to find the domain. Well first you've got to make sure you know what domain means. So domain is the set of all numbers X. That you can put into this function so that a real number comes back out again. So is there any way you could put a number in there? A real number? So that an imaginary number comes out something with? I. No. Okay so that's not a problem. The problem is what if you put a number in and you got this? So some number over 0? Well some number over zero is undefined. That's not a number. Okay. So if there are any numbers that caused the bottom to be zero then they are not in the domain of this function. And if you think about graphing, if it makes the bottom zero then there's a vertical sm tote there. Okay and that's a that's a line that the graph can't cross. So to find the domain of this function you have to find out what X. Is. Make the denominator zero and then leave those out. Okay. Like this. Yeah. So I bet I can factor the bottom. If I can't I have to use the quadratic formula on it. Um Two X x minus one plus one. That gives me two X squared minus two X plus x. Okay that'll work. Okay. two x plus one X minus one. You can either say can't equal zero here or equal zero here because you're going to use the information in a minute. So X equals want and X equals minus one half. So domain is all real numbers except X equals minus one half And x equals one. So as long as you don't put those two numbers in there then this function is fine. But if you put those one or the other of those in there then this function causes this undefined thing to happen and so that those two things are not in the domain. Okay. Here's another way to write it because I'm sure it doesn't look like that in your book and it's easy if you put it on a number line. So it's all the numbers starting at minus infinity Up to -1/2 but don't include it together with so I'm gonna put a union sign there All the numbers on this side of -1 half. Up to the numbers up to the # one, so minus one half to one Together with all the numbers on this side of one up to positive infinity. Okay. Got says leave out minus one half and one because the round bracket means don't include. Okay, so this and this say exactly the same thing. All right. Now, why is this function continuous at every point in its domain. Okay, um let's look at this one. two x plus one is a line, X -1 is a line. Those two are continuous. When you multiply continuous functions you get a continuous function. Yeah, X squared plus one is a parabola that's continuous. When you divide a continuous function by a continuous function, it's continuous. So this function is continuous everywhere there is defined. Okay, so the rules I used, I don't know what numbers they are in your book, but that product product of continuous functions is continuous and quotient of continuous functions is continuous and um the fact that each of these three are continuous because they are, I don't know if you have one that's Paulino meals are continuous. Okay, hope that help.

Get More Help with this Textbook

James Stewart

Calculus: Early Transcendentals

View More Answers From This Book

Find Another Textbook

Study Groups

Study with other students and unlock Numerade solutions for free.

Problem