Determine the interval(s) on which the following functions are...

Question

Determine the interval(s) on which the following functions are continuous. At which finite endpoints of the intervals of continuity is f continuous from the left or continuous from the right?

f(t)=(t^2−1)^3/2

Accepted Answer

Hi everyone, let's take a look at this practice problem dealing with continuity. This problem says find the interval for the continuity of the function. Also identify the endpoints for which the function is left continuous or right continuous. And we're getting the function G of X is equal to the quantity of X2 minus 121 in quantity raised to the 3 divided by 2. We're given 4 possible choices as our answers. For choice A, we have the left open interval of minus infinity, to the right closed interval of -11, and it's left continuous at -11. For choice B, we have the left closed interval of 11, 2, the right open interval of infinity, and it's right continuous at 11. For choice C, we have the left open interval of minus infinity to the closed right interval of -11, union with the left close interval of 11 to the right open interval of infinity, and we have it being right continuous at 11 and left continuous at -11. For choice D, we have the left open interval of minus infinity to the right closed interval of -11. Union with the left close interval of 11 to the right open interval of infinity, and it's right continuous at -11 and left continuous at 11. Now, in order to determine the interval of continuity, we first need to determine the domain over which this function is valid. And here, we actually have a square root in the form of a power raise to the 1/2. So we need the function that's inside of the square root term to be greater than 0. That means that X2d. -121. Has to be greater than or equal to 0. Now, I'll need to solve this um expression for X, and so to do so, I'll first add 121 to both sides. Then I'll have X2d is greater than or equal to 121. I can then take the square root of both sides. Solve for X, and the square root of X2 is going to be the absolute value of X. And this could be greater than or equal to the square root of 121, which is 11. That means that X can be greater than or equal to 11, and X is going to be less than or equal to minus 11. So, to write that as an interval. That is going to be the left open interval, going from minus infinity. To the right close interval of -11. Union with the left closed interval of 11 to the right open interval of infinity. And so that's gonna be the domain for my function, and since I have a even polynomial inside of a square root function, that means that our function is going to be continuous. Over the entire domain of the function. So, while this is our domain for our function, this is also the interval of our continuity for our function. So now that we have the um intervals over which the function is continuous, we now need to look at the end points. So here we're gonna need to look at plus and minus 11. So, we'll start off with X equal to -11 1st. And we need to determine whether this is left continuous or right continuous. In this case, it is going to be left continuous. That's because the function is defined and continuous. To the left of X equal to -11 as it approaches X equal to -11 from the left. Likewise, if we look at X equal to 11, it is going to be right continuous. That's because the function is continuous and defined to the right of X equal to 11 as it is approaching X equal to 11. So now we've determined the continuity for our two endpoints. We have enough information to look at our answer choices. And the correct answer choice actually corresponds to answer choice D. So I hope that this explanation has been useful, and I'll see you in the next video.

Determine the interval(s) on which the following functions are continuous. At which finite endpoints of the intervals of continuity is f continuous from the left or continuous from the right?

f(t)=(t^2−1)^3/2

Key Concepts

Continuity of Functions

Endpoints of Intervals

Piecewise Functions and Domain Restrictions

Watch next