Hole Asymptote or continuous?

The video tutorial explains how to identify discontinuities in rational functions. It begins by introducing the concept of discontinuities and the importance of showing work in free response problems. The instructor discusses the challenges of factoring and explains that the given function cannot be factored under real numbers. The process of finding discontinuities involves setting the denominator to zero and solving, but in this case, no real solutions exist, indicating no discontinuities. The tutorial concludes by affirming that the domain is all real numbers, with no holes or asymptotes.

What is the function presented in the problem to explore discontinuities?

• f(x) = x^2 + 3x + 1

• f(x) = (3x + 1) / x^2

• f(x) = x^2 / (3x + 1)

• f(x) = 3x / (x^2 + 1)

What is the first step in identifying discontinuities in a rational function?

• Factor the denominator

• Factor the numerator

• Set the denominator equal to zero

• Set the numerator equal to zero

Why can't the equation x^2 + 1 = 0 be solved for real numbers?

• It results in an undefined number

• It results in a zero

• It results in a negative number

• It results in a complex number

What does the absence of real solutions for x^2 + 1 = 0 imply about the function's domain?

• The domain is all complex numbers

• The domain is all real numbers

• The domain is all negative numbers

• The domain is all positive numbers

What conclusion is drawn about the function's discontinuities?

• There are non-removable discontinuities

• There are removable discontinuities

• There are infinite discontinuities

• There are no discontinuities