Question

1. - / 6.25 points explain, using the theorems, why the function is continuous at every number in its domain. $f(x)=\frac{2x^{2}-x - 1}{x^{2}+1}$ $f(x)$ is a polynomial, so it is continuous at every number in its domain. $f(x)$ is a rational function, so it is continuous at every number in its domain. $f(x)$ is a composition of functions that are continuous for all real numbers, so it is continuous at every number in its domain. $f(x)$ is not continuous at every number in its domain. none of these state the domain. (enter your answer using interval notation.) resources read it

Explanation:

Step1: Identify function type

The function $F(x)=\frac{2x^{2}-x - 1}{x^{2}+1}$ is a rational function since it is in the form $\frac{P(x)}{Q(x)}$ where $P(x)=2x^{2}-x - 1$ and $Q(x)=x^{2}+1$ are polynomials.

Step2: Recall continuity of rational functions

A rational function $\frac{P(x)}{Q(x)}$ is continuous at every number in its domain. The domain of a rational function is all real - numbers except the values of $x$ that make the denominator $Q(x) = 0$.

Step3: Find the domain

For $Q(x)=x^{2}+1$, set $x^{2}+1 = 0$. Then $x^{2}=-1$. Since there are no real - numbers $x$ such that $x^{2}=-1$, the domain of $F(x)$ is all real numbers, which in interval notation is $(-\infty,\infty)$.

Answer:

1. B. $F(x)$ is a rational function, so it is continuous at every number in its domain.
2. $(-\infty,\infty)$