continuity in interval notation (explain how you get the notation)

[calm depot]

I'm trying to study for the final exam but the answer key doesn't explain how they got the given interval notation. I might've forgotten some type of rules, could someone explain. Thank you.

[eager shell]

do you understand the question? if so i think you should know how to answer at least one of them

[nocturne dragon]

if a function is continuous along some interval [a,b] or (a,b), then it means that as you range from x = a to x = b, the function is continuous on all spots essentially

i could give a rigorous definition of continuity but that doesn't seem needed here

for example, in (a), that polynomial is continuous for all values of x, so it is continuous on (-infty, infty)

for (b), f(x) is not continuous at -2 and 2. so, it's continuous from -infinity to -2, (-infty, -2), using an open interval since we want to disclude -2
it's then continuous from -2 up to 2 again, not including 2 and -2, so we write (-2,2)
it's continuous again from 2 to infinity, so (2, infinity)

so you just take the union of those 3 intervals

[robust haven]

do you know why it’s not continuous at those values? that’ll help you do these questions a lot!! (:

[robust haven]

you dingus! they were supposed to answer that. ):<

[hot drum]

I thought you were them

thought there were only 2 people in the convo, not 3

[robust haven]

heh no worries! i’m only joking with you. (:

just want to make sure they know what they’re doing without being told. o:

[hot drum]

good, I was about to contact my lawyer

[robust haven]

your lawyer is a dingus too.

[hot drum]

[nocturne dragon]

yeah, i'll explain when i get to school lmao

[nocturne dragon]

for (b), f(x) isn't continuous at -2 and 2 because the denominator x^2-4 = 0 at -2 and 2

for (c), sqrt(x+1) isn't defined if x+1 < 0 => x < -1
so x must be > or = -1
the bottom (denominator) is just x, so the function isn't defined for x = 0

i can explain the others if you want

just lemme know

[robust haven]

well.. it seems like you know what you’re talking about.

so what do you need help with? o:

do you just want to know how to write it in interval notation orrr…? o:

[nocturne dragon]

oh nvm sorry

i thought you were the original question asker lol

😭

[robust haven]

LMAOOO WHY DID I THINK THE SAME THING. 😭

you weren’t the first one to do that!!! o:

whoever the person is that actually needs help i’m sorry, we are a messsss LMAO.

[nocturne dragon]

lmaooooo

[nocturne dragon]

someone who needed help probably wouldn't have said this 💀💀

[robust haven]

yeah i was sitting there like what the fuck why does this person need help. 😭😭

thought that maybe they were trying to flaunt their knowledge. 🥹

[nocturne dragon]

BAHAHAHA