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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} $

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

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Missouri State University

Harvey Mudd College

Baylor University

University of Nottingham

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.

Oklahoma State University