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(x)=√25−x^2

Accepted Answer

Hi everyone, let's take a look at this practice problem dealing with continuity. This problem says find the interval of continuity for the function. Also identify the endpoints for which the function is left continuous or right continuous. When we're given the function, G of X is equal to the square root of the quantity of 289 minus X squared. We're given 4 possible choices as our answers. For choice A, we have the closed interval of -289 to 289, and that is left continuous at -289 and right continuous at 289. For choice B, we have the closed interval of -289 to 289, and it is left continuous at 289 and right continuous at -289. For choice C, we have the closed interval of -17 to 17, and it is left continuous at -17 and right continuous at 17. And for choice D, we have the closed interval of -17 to 17, and it is the left continuous at 17 and right continuous at -17. Now, we're asked to find the interval of continuity for this function, and the interval of continuity is going to be the same as the domain of our function, since we have a polynomial here. However, we need to look at um where our function is actually defined, and our function is only defined when the polynomial underneath the square root sign is greater than or equal to 0. So that means that 289. Minus X2 has to be greater than or equal to 0. So, I actually need to solve this for X to determine where my um end points are. And so the first step I'm going to do is add X2d to both sides. In doing so, I will get 289 is greater than or equal to X2, and now I'm going to take the square root of both sides. In doing so, the square root of 289 is 17, and so that has to be greater than or equal to, and when I take the square root of X2, that gives me the absolute value of X. So, this gives me two endpoints, so I can have X is less than or equal to 17. Or I can have X is greater than or equal to -17. So that means that the domain of my function is the closed interval from -17 to 17. That's also going to be my interval of continuity. That's the first part answer for this question. Now we need to identify the end points for which the function is left continuous or right continuous. So here we need to look at our two endpoints. That's going to be X equal to -17 and X equal to 17. So we're first going to look at X equal to -17. And this end point is actually right continuous. And that's because the function is defined and continuous. To the right of -17. Likewise, if we look at X equal to 17, that is going to be left continuous. Because our function is defined and continuous. From the left of X equal to 17. So now we've identified the continuities for our two endpoints, as well as our interval of continuity, we can now look at our answer choices, and the correct answer choice 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(x)=√25−x^2

Key Concepts

Continuity of Functions

Endpoints of Intervals

Square Root Function

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