Question

using the appropriate theorems, explain why the function is continuous at every number in its domain.
f(x) = \frac{2x^2 - x - 4}{x^2 + 1}
\bigcirc f(x) is a polynomial, so it is continuous at every number in its domain.
\bigcirc f(x) is a rational function, so it is continuous at every number in its domain.
\bigcirc f(x) is a composition of functions that are continuous for all real numbers, so it is continuous at every number in its domain.
\bigcirc f(x) is not continuous at every number in its domain.
\bigcirc none of these
find the domain. (enter your answer using interval notation.)

Explanation:

Part 1: Choosing the Correct Reason for Continuity

A rational function is defined as \( \frac{f(x)}{g(x)} \) where \( f(x) \) and \( g(x) \) are polynomials and \( g(x)
eq 0 \). For \( F(x)=\frac{2x^{2}-x - 4}{x^{2}+1} \), the numerator \( 2x^{2}-x - 4 \) and denominator \( x^{2}+1 \) are polynomials. A polynomial is continuous everywhere, and for a rational function, it is continuous at all points in its domain (where the denominator is non - zero). The denominator \( x^{2}+1 \) has no real roots since \( x^{2}+1 = 0\) implies \( x^{2}=- 1\), which has no real solutions. So \( F(x) \) is a rational function, and rational functions are continuous at every number in their domain.

Step 1: Analyze the denominator

The function \( F(x)=\frac{2x^{2}-x - 4}{x^{2}+1} \) is a rational function. The domain of a rational function is all real numbers except the values of \( x \) that make the denominator equal to zero. We set the denominator equal to zero and solve for \( x \):
\( x^{2}+1 = 0\)

Step 2: Solve the equation for \( x \)

For the equation \( x^{2}+1 = 0\), we can rewrite it as \( x^{2}=-1\). In the set of real numbers, there is no real number \( x \) such that \( x^{2}=-1\) (because the square of any real number is non - negative). So the denominator \( x^{2}+1 \) is never zero for any real number \( x \).

Step 3: Determine the domain

Since there are no real numbers that make the denominator zero, the domain of the function \( F(x) \) is all real numbers. In interval notation, the set of all real numbers is written as \( (-\infty,\infty) \).

Answer:

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

Part 2: Finding the Domain