Question
(7]Where is the function f (2) continuous? Find and classify all discontinuities_COSE (I - 3)if z 2 3 if I < 3 ~5) - 1)f(z)
(7] Where is the function f (2) continuous? Find and classify all discontinuities_ COSE (I - 3) if z 2 3 if I < 3 ~5) - 1) f(z)
Answers
At what values of $x$ is the function not continuous? If possible, give a value for the function at each point of discontinuity so the function is continuous everywhere. $$f(z)=\frac{z^{2}-11 z+18}{2 z-18}$$
In this problem, we are given the function F f x is equal to negative three X cubed plus seven and were required to find the numbers at which F is continuous and discontinuous. Now we can note that this function is a cubic polynomial, so it has only individual powers effects and the highest industrial power is three. And we know that all polynomial are continuous functions. So from that we can conclude that ffx is continuous for all real values effects. And that implies that ffx is discontinued, so discontinuous for no real value effects. So this function is continuous everywhere, and it's discontinuous nowhere. And that is the solution to this problem.
Let's find a value for A. So that the function is continuous. So we have z squared minus one. And this is gonna be the case as X approaches three from the left. So the value that's being approached is three squared which is nine minus one, which is eight. So eight is the value that we're approaching and we know that we have to a X. We're approaching three from the right now. So if we plug in a three here for X that end up giving us six a And we wanted to equal eight. So in order to ensure that we do a equals eight divided by six, Which is going to be 4/3. So a equaling 4/3 would be our final answer. Um And moving on into a different realm, we have E to the negative X equaling X. We know that this is only going to have one solution and we already know that one of the solutions is going to be the fact that when we subtract this, we're not getting one solution right here and that's because this function is going to if we draft them separately, we see that this is just going to go like this or is this function is continuously decreasing. So there's never going to be a point where they cross the second time
Our goal is to determine whether or not the function that's given to us as continuous. So in this case it's going to be F as a function of X is equal to 21 minus seven X over back to my next three. So zooming out, we see that is just going to be a straight line. Um Then looking at the limit as Acts approaches three from the left and the right, we see that it's going to be negative seven, so this is equal to negative seven. However, we see that when we evaluate it F of three, it's undefined. So therefore this is not going to be continuous for this very reason. Yeah.
Mhm. Okay, so our function F of X was equal to the absolute value of x minus three. Over x minus three can be written as fx is either equal to negative one if x minus three is less than zero. So that implies that X is less than three or x minus three. Um over X -3. Right? If it's eager to one and that's if x minus three is greater than or equal to zero, which implies that X is greater than or equal to three. So since F of X is defined for all X, not equal to three, we have to check continuity. I'm only at X is equal to three. But if we take the limit as X approaches three from the left, we get negative one. And the limit as X approaches three from the right we get one. So therefore the limit as X approaches three does not exist. And since um fx can be modified as a continuous function at X is equal to three. Therefore ffx has a non removable. This continuity. Um at X is equal to three.