Problem

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

View

Problem 25 Easy Difficulty

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

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

Answer

$F(x)=\frac{2 x^{2}-x-1}{x^{2}+1}$ is a rational function, so it is continuous on its domain, $(-\infty, \infty),$ by Theorem $5(b)$

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

So in this problem were given this function F of X equals two X squared -X -1. Over X squared plus one. Okay. Rescue stems 457 and nine. To explain why this function is continuous over its domain. And then and then list the domain. Okay, well, first of all we can think of this function as two functions H of X divided by G F X. Where H of X. Is this polynomial right? Two X squared minus x minus one. Okay. And one of our theorems says that polynomial is continuous over all real numbers. Okay, So then G F x is x squared plus one, which by the same theorem this is a polynomial. So it is continuous over all real numbers. I also want to note that G F X is never zero. Right? No matter what I put in here for X, Even if I put zero in there square zero, you get zero. But then you add one to it. So it's never zero anytime you square a number, you're gonna get some positive number and then you're going to add one to it. So you're always gonna have something that's never zero. Okay, so then by the by the other thing we have F of X is H X over G X. Where GFX is not zero, then f of X is a product uh continuous functions. So by another term we have there ffx is continuous. Okay. Now, since I can put all real numbers in two G fx And I never get zero and I can put all real numbers into H of X as well. Then that means that the domain of F of X is all real numbers, in other words, from minus infinity to infinity. Okay, So there you go. That is by those terms, that's why F of X is continuous and the domain is all real numbers.

David M.

Oklahoma State University

Related Courses

Calculus 1 / AB

Calculus: Early Transcendentals

Chapter 2

Limits and Derivatives

Section 5

Continuity

Problem