Complete the following steps for each function.

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Question

Complete the following steps for each function.

c. State the interval(s) of continuity.

f(x)={2x if x<1

x^2+3x if x≥1; a=1

Accepted Answer

Hi, everyone. Let's take a look at this practice problem dealing with continuity. This problem says determine the intervals of continuity for the piecewise function G of X defined as follows. We have G of X is equal to 3X minus 2 if X is less than or equal to 2, and X cubed minus 4 X if X is greater than 2. We're given 4 possible choices as our answers. For choice A, we have the open interval of minus infinity D to 2. For choice B, we have the open interval of 2 to infinity. For choice C, we have the open interval of minus infinity 2 minus 2 union with the open interval from 2 to infinity. And for choice D, we have the open interval from minus infinity to 2, union with the open interval 2 to infinity. Now, the question asks us to determine the intervals of continuity for a piece-wise function, and so what we're going to do is look at each piece independently. And so for our first piece. We actually have the function 3X minus 2, and this is 4 x less than or equal to 2. So we're going to ignore the endpoint at the moment and look at the case of X strictly less than 2. We're going to look at X equal to 2 as a separate case. And so for X, less than 2, we have a linear function, which is continuous everywhere. In its domain And so, we can write that that interval of continuity for our linear function here for the first piece, as the open interval from minus infinity to 2. We can look at the next piece, which is our polynomial, which is XQ minus 4 X. And 4 X greater than 2, we'll be using that piece, and a polynomial, in this case, is continuous everywhere in this domain. And so, the interval of continuity for that piece is going to be the open interval from 2 to infinity. So now the only case that we need to look at is when X is equal to 2. And for this, we're going to apply our checklists for um our continuity for a single point. And so recall for that checklist, we have three conditions that need to be met. The first condition is that F of A is defined, so the function has to be defined at the point. For the second condition, we need the limit as X approaches A of F of X. To exist, the limit must exist. And for our third condition, we need the limit as X approaches A of F of X. 2 equal f of a. So the limit as X purchase A of our function has to be equal to the value of the function at that point. So, we need to basically look at these 3 conditions. So, for the first condition, we need to see that FFA is defined. So, in our case here, we're going to be looking at GF2. And GF2 is actually covered in our first piece, since X can be less than or equal to 2, and so in that case GF2 is defined. So our first condition. I met So now we're going to look at the second condition. And so we need to see if the limit as X approach approaches A of F of X exists. So we call that for a limit to exist. The limit must be the same value whether I approach the value from the left or right side of the function. So we're gonna be looking at two different limits. First, we'll be looking at the limit as X approaches. 2 from the left of G of X, and that means X is going to be less than 2, so we'll be using our first piece, which is going to be 3 X minus 2. And so when I evaluate this limit, this is going to be equal to 3 multiplied by 2 minus 2, which is going to be equal to 4. So, now we need to evaluate the limit, as X approaches 2 from the right. So, for that, we'll have the limit as X approaches 2 from the right. And since I'm coming from the right, that means X is going to be slightly greater than 2, so we're gonna be using our second piece, which is going to be X cubed minus 4 X. And so evaluating this limit, it's going to be equal to 2 cubed minus 4 multiplied by 2, which is equal to 0. So here, I have two limits, but their values are not the same. And in order for the limit to exist, they both must be the same. So, our second condition is not met. And I do not need to check the last condition, because if any of our conditions are not met, then the function is not continuous at this point. So, our function is continuous everywhere except for the point X equal to 2. And the way we can write that as in interval notation, that it's just going to be the open interval from minus infinity to 2, which was our um interval of continuity for our first piece, Union. With the open interval of 2 to infinity, which was our interval of continuity for our second piece. And notice that since I have an open interval um at the point X equal to 2, X equals 2 is not um contained in our interval of continuity. So this is the the answer that I'm looking for. And if I come over and look at my answer choices, 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.

Complete the following steps for each function.

c. State the interval(s) of continuity.

f(x)={2x if x<1
x^2+3x if x≥1; a=1

Key Concepts

Continuity of Functions

Piecewise Functions

Limits

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