At what points are the functions in Exercises 13–30 continuous...

Question

At what points are the functions in Exercises 13–30 continuous?

y = √(x⁴ +1)/(1 + sin² x)

Accepted Answer

Welcome back, everyone. Determine the points at which the function G of X equals square root of X2 + 9 divided by 1 + cosine square of X is continuous. We're given four answer choices A says all X, B X X equals -3, C X X equals 3, and D all X except X equals -1. So let's analyze the given function where given G of X equals square root of X2 + 9 divided by 1 plus cosine squared X. If we want to identify where G of X is continuous, we have to identify its domain. It is going to be continuous in the domain. And because G of X is an irrational function, we have to recall that division by zero is undefined. So we want to make sure that our denominator, which is 1 plus cosine quad of x, is not equal to 0, right, because in this case we will be dividing by 0, which will be not defined. And we will have to exclude all of the X values that make the denominator 4 to 0 to see where the function is continuous. Let's recall that the range of a cosine function. Is between -1 and 1, right? Inclusive. Now in this case we have cosine squared of X. What does that mean? Well, if we consider cosine squared off X, the minimum value that we can get is actually 0, right? If we have a square of a number, it cannot be negative. The minimum value that we can take is 0, right, because the minimum value of a square of some number is 0, and the maximum value is going to be 1. We're basically squaring the upper bound to get 1 or the lower bound if let's say cosine reaches its extreme value of -1, -1 squared is 1. So essentially we can sell that cosine squared of X is between 0 inclusive and 1 inclusive. So if we take 1 plus cosine squared X, It simply means that we're going to add 1 to all sides of the inequality, so 1 plus cosquad of X is between 1 and 2 inclusive, right? And what can we see? Well, it is never equal to 0, right, because essentially we have values between 1 and 2 for 1 plus cosine square x. So our denominator is never lower than 1 and therefore it cannot be equal to 0 as a result, meaning the The domain for this function is all real numbers. We can say that the domain is X belongs to all real numbers because the denominator is number equal to 0 for any X and therefore, our function is continuous for all X. Thank you for watching.

At what points are the functions in Exercises 13–30 continuous?
y = √(x⁴ +1)/(1 + sin² x)

Key Concepts

Continuity of Functions

Domain of a Function

Behavior of Square Roots and Trigonometric Functions

Watch next