help-23

[rustic goblet]

what's the range of f(x)?

[midnight mirage]

ok so range is like the highest point and the lowest point

[rustic goblet]

the range of f(x) is the set of all y values it takes

[midnight mirage]

if u look at the graphn, highest point is starting infinitely and ends infnitely

but goes down so

[rustic goblet]

!

[midnight mirage]

(inifinity, -inifinity)

[rustic goblet]

you mean (-inf, inf)?

[midnight mirage]

yeah sorry

[rustic goblet]

[midnight mirage]

bc it goes down and comes up!

[rustic goblet]

yeah, that's right (though you can write it as just R)

R for the real numbers

(-inf, inf) = R

[midnight mirage]

yeah but in this one i will have to write it in that form

[rustic goblet]

yeah, I noticed

(-inf, inf) is fine

okay, part c)

what're the x-intercepts?

[midnight mirage]

ok x intercepts is when it touche the X axis

[rustic goblet]

yes

[midnight mirage]

so -2,2,4

[rustic goblet]

!

good

well...

not quite

[midnight mirage]

do the signes get switched?

[rustic goblet]

we want ordered pairs

not just the x coordinate

[midnight mirage]

ok so then that would be,

-2,0 2,0 4,0

[celest slate]

(x,y)

[midnight mirage]

yup

[rustic goblet]

(-2,0), (2,0), (4,0)

[midnight mirage]

0 is the wy

[rustic goblet]

but yes

[midnight mirage]

y

[rustic goblet]

you shouldn't forget the parentheses

they are important

[midnight mirage]

yup!

[rustic goblet]

otherwise, it looks like you're just writing numbers

not ordered pairs

okay, part d)

an easy one

the zeros are just the x coordinates of the x intercepts

[midnight mirage]

ok correct me if im wrong right

[midnight mirage]

this will just be switched so the zeros are gonna be, 2,-2,-4

[rustic goblet]

what do you mean?

[midnight mirage]

the sign will be switched

if the x cordinate is 2 its 0 will be -2 right?

[rustic goblet]

that is not correct

[midnight mirage]

oh!

[rustic goblet]

why would the sign switch?

at x = -2, f(x) = 0

[midnight mirage]

ahhhh

[rustic goblet]

so the x intercept is (-2, 0)

same for x = 2 and x = 4

[midnight mirage]

ok 0's are just the x cordinates of the x intercpets

[rustic goblet]

no

[midnight mirage]

ohh i get it

[rustic goblet]

the 0's are the value that f(x) takes on at x

the ordered pairs are (x, f(x))

when x = -2, we have the ordered pair (-2, f(-2))

but f(-2) = 0

so it becomes (-2, 0)

[celest slate]

y=f(x),
(x,y)=(x,f(x))

[midnight mirage]

OOO

[rustic goblet]

indeed

[midnight mirage]

-2,2,4 are the zeros!

[rustic goblet]

they are!

[midnight mirage]

got that :D

[rustic goblet]

how's part e)?

what is f(-3)?

[midnight mirage]

now e

f (-3) is (x,f(-3)

[rustic goblet]

oh no

[midnight mirage]

ohhhh

[rustic goblet]

f(x) does not output an ordered pair

it outputs a single real number

we can't say that f(x) = (x, y)

that isn't the output of f(x)

[celest slate]

my pc is flashbanging i cant see the values xddd

[rustic goblet]

(side note: you can have functions that map a single number to an ordered pair, but we don't have that here)

we have a function that maps a real number to a real number

[midnight mirage]

wait i think i got it the answer should be 5

[rustic goblet]

!!

good

f)?

[midnight mirage]

and the f - 2 should be

[rustic goblet]

you can do it!

[midnight mirage]

wait what the frick

-4? or 0

one of it

[rustic goblet]

[midnight mirage]

its 0

[celest slate]

[midnight mirage]

got that

[rustic goblet]

[midnight mirage]

LETS GOOO

[rustic goblet]

okay, now g)

what is f(0)?

[midnight mirage]

you guys are actually better than my teachers

thats easy -4

[rustic goblet]

good

[midnight mirage]

so its negative

[celest slate]

because students understand eachother the best!

[rustic goblet]

good!

okay, now h)

how many times does y = 2 intersect f(x)?

[midnight mirage]

so i know that it intercets once at x = 3

[rustic goblet]

mhm

where else?

[midnight mirage]

when x = -2.5

[rustic goblet]

right

[midnight mirage]

i think, i might be fully wrong tho

[rustic goblet]

you are correct

[midnight mirage]

yay

this is so easy

[midnight mirage]

thank you guys so much ive been stressing it the whole day

[rustic goblet]

two intersections

[rustic goblet]

don't worry about it

[midnight mirage]

yup!

[rustic goblet]

I'm happy to help!

[midnight mirage]

ok now I

[rustic goblet]

so when does f(x) = 5?

[midnight mirage]

at -3 !

[rustic goblet]

right.

anywhere else?

[midnight mirage]

nope

just once i tink

think

[rustic goblet]

good

two more to go!

onto j)

when is f(x) positive?

[midnight mirage]

at 3

x = 3

[rustic goblet]

sure, but on what intervals is f(x) positive?

[midnight mirage]

(3,2)

ok wait i actually dont know how to do j and k

[rustic goblet]

okay

that's okay

let's do it together

[midnight mirage]

okay!

[rustic goblet]

can you find the points where f(x) is 0?

[midnight mirage]

yeah when y = -4 f(x) is 0

[rustic goblet]

hmm, that's not quite what I was asking for

here, look at the part I've shaded blue

is f(x) on that part positive?

[midnight mirage]

ohh

yes its possible in -2

[rustic goblet]

well... no

f(-2) = 0 at x = -2

[midnight mirage]

oh

[rustic goblet]

0 is not positive

but for all x < -2, is f(x) > 0?

this isn't a trick question haha

[midnight mirage]

ok wait

[midnight mirage]

should be right?

WAIT NO

it cant

because -2 is less than 0

[rustic goblet]

well, -2 is the input

we're asking when the output is positive!

that's very different

[midnight mirage]

OHH

[rustic goblet]

the output is f(x)

so when is the output > 0?

[midnight mirage]

the output wil be positve when f(x) is -2

[rustic goblet]

nono

[midnight mirage]

oh :(

[rustic goblet]

that's wrong in multiple ways

first off, f(x) is not -2, x is -2

[midnight mirage]

mhm

[rustic goblet]

you're conflating the input and the output >.<

secondly, we just saw in part d) that x = -2 is a zero of f(x)

that means that f(-2) = 0

which is not positive

[midnight mirage]

OHHHH

i see what u mean now i get it

kinda but i think

[rustic goblet]

I'm not asking you to look at individual inputs

I'm asking you to look at a whole breadth of inputs

and find where the output is positive

[midnight mirage]

ok so the graph is comming down so -infinity then hits X axis -2

[rustic goblet]

yes!

it's coming down

and then hits the x-axis

[midnight mirage]

and then it goes down to -4 then it goes up

[rustic goblet]

indeed

[midnight mirage]

so our positive starting point would be -4

[rustic goblet]

tell me, when x < -2, is f(x) > 0?

[midnight mirage]

and then it hits 2 and goes down at 4

[rustic goblet]

no >.<

-4 has nothing to do with this

[midnight mirage]

oh

[rustic goblet]

hmm, maybe I can ask you in a different way

is f(-100) positive?

[midnight mirage]

nope its negative right?

[rustic goblet]

why is it negative?

the input, x = -100, is negative

but what about the output, f(x)?

[midnight mirage]

infinity?

[rustic goblet]

it's off the page, but it's not infinity

infinity is not a real number

real valued functions cannot ouput infinity

I'm only asking if f(-100) is positive

not what value it is

[midnight mirage]

its negative because of the sign? 💀

[rustic goblet]

what sign?

[midnight mirage]

the neative sign

[rustic goblet]

.

you're conflating input and output

[midnight mirage]

OHH

you want me to find the output

[rustic goblet]

yes, is the output a positive number?

I do not need you to find it exactly

[midnight mirage]

theres no way to know tho is there?

[rustic goblet]

you're right, but just by judging, what do you think?

(in hindsight, x = -100 was probably not the best choice on my part)

what about f(-4)?

is it positive?

[midnight mirage]

the f(x) can be positive I think because eventually the graph will go at -4

[rustic goblet]

I do not understand

is f(-4) positive or not?

it's either a yes or a no, not "it can be"

[midnight mirage]

is it possible if u can explain me why its positive or negative?

maybe then i can learn

[rustic goblet]

sure

[midnight mirage]

and ill use that logic to answer your other question, if f(-2) is it positive or negative

[midnight mirage]

yeah this is what I meant by this

[rustic goblet]

at x = -4, the graph looks like it's about at 11

[midnight mirage]

but i wasnt sure

[rustic goblet]

11 is positive, so f(-4) is positive

what about f(-3)?

[midnight mirage]

-3 will be positive around 4.7 or 4.8

[rustic goblet]

good

what about f(-2.2)?

[midnight mirage]

which means that 2.2 will be positive around 0.3-0.4

[rustic goblet]

yes

what about f(-2)?

[midnight mirage]

and 2 is positive at 0

it cant be negative

[rustic goblet]

0 is not a positive number

[midnight mirage]

so its neutral

[rustic goblet]

it is neither positive nor negative

yes

so for all x < -2, we have f(x) > 0

does that make sense?

[midnight mirage]

YUP!

[rustic goblet]

okay, what about f(2.1)?

positive?

[midnight mirage]

yes but it will be a very very small number , 0.2 or 0.1

[rustic goblet]

right

I'm not interested in how small the number is

I only care if it is a positive number or not

[midnight mirage]

so yes its positive

[rustic goblet]

we can repeat this to see that f(3), f(3.9), etc are positive

[midnight mirage]

yes

[rustic goblet]

so for all x between 2 and 4, f(x) > 0

the graph is above the x-axis for all x between 2 and 4

so f(x) is positive for all x between 2 and 4

[midnight mirage]

yeah

[rustic goblet]

that is the quick way to see it

[midnight mirage]

yes

so thats another 2 key points?

wait so would that be

[rustic goblet]

so f(x) > 0 when x < -2 and when 2 < x < 4

[midnight mirage]

(-inifnity , -2) U (2,4) ?

[rustic goblet]

yes!

[midnight mirage]

because 2, 4 is when tis positive and -2 is when its neutral but its commming down

[rustic goblet]

sure

[celest slate]

and do you know why we dont include 2 and 4 with a closed bracket?

why is it (2,4) and not [2,4]

[midnight mirage]

isnt it because 2,4 are set points?

[celest slate]

not quite, what are 2 and 4 equal to again when plugged in?

[midnight mirage]

OH BEAUSE THEY ARE 0's

when plugged in

[celest slate]

exactly!

and 0 is not positive

[midnight mirage]

is that why we dont use [ for -2 as well?

[celest slate]

mhm

[midnight mirage]

because -2 is - as well when plugged in

[celest slate]

it is 0 aswell and so its not a positive value

so we dont include it and we use an open bracket

[midnight mirage]

wow

man i love math when i understand things 😂

[celest slate]

realll

🔥

[midnight mirage]

ok no K

ow

now

[celest slate]

now k yeah

[midnight mirage]

ok now so its asking when f (x) is LESS THAN 0

[celest slate]

where is f(x)<0

[midnight mirage]

which is between -2 and 2

thats when the graph is at negative

[celest slate]

yes

anywhere else?

[midnight mirage]

yup!

after 4

but then it goes down infinitely and dosent touch the verticle assympotppe

[celest slate]

yes

[midnight mirage]

(-2,2)U(4,-infinity) ?

[celest slate]

but again, since it doesnt touch the asymptote doesnt that mean that we have to exclude 6?

as it doesnt have a value

[midnight mirage]

we dont

[celest slate]

im not quite sure about that

[rustic goblet]

the function's domain does not include 6

it takes on no value there, so how can it be negative?

-infinity is not negative, because it is not a real number

[midnight mirage]

so

(-2,2)U(4,infinity) ?

[celest slate]

refer to this @midnight mirage

[celest slate]

.

[midnight mirage]

ah the only reason I said - inf is becauyse it goes down after 4

[celest slate]

no no thats not the issue, you fixed the inf but the problem here is that when you say (4,inf) you are including 6 which is NOT a negative value

[midnight mirage]

ah

[celest slate]

how do you think we can type it in a way that makes us exclude 6 from (4,inf)?

[midnight mirage]

(]

like this so it cloes

closes

[celest slate]

not really

[midnight mirage]

oh then im lost sorry

[celest slate]

if i give you a set like;
(2,10)-{7} what do you understand from this?

[midnight mirage]

i get that 2,10 is one whole cordinate and the it goes to 7

i dont know what the parenthesis does

[celest slate]

this basically means the numbers 3,4,5,6,8,9 which are the numbers between 2 to 10 excluding 7

[celest slate]

the paranthesis determines if we include the value at the end or beginning or not

[celest slate]

take this example

if we had written the set like;
[2,10]-{7} the values it can take would be 2,3,4,5,6,8,9,10

notice the difference?

[midnight mirage]

just not the 7 because its excluded right

[celest slate]

yeah exactly, when we wrote it with closed brackets so [] we included the number at the beginning and the end

[midnight mirage]

ah so my answer would be

[celest slate]

but when we wrote it with () we didnt include them

[midnight mirage]

(-2,2)U{4,-infinity]

[celest slate]

when you type it like this that means -infinity is included which is not possible, infinity is not a number

also its not negative infinity

[midnight mirage]

oh

[celest slate]

as its going to the positive side

which means its positive

[midnight mirage]

i am lost kinda again, can you reverse it like say the answer and why its that

[celest slate]

so

basically

(-2,2) is the first part, we put them in () and not in [] because -2 and 2 do not give a negative value they give 0.
the second part of the answer is (4,6), the reason for that is because the asymptote is like a void, you mentioned that the graph doesnt touch the asymptote which means at 6 which is where the asymptote is there is no value so its not negative and we cant include it which is why we type it in (). now the last part is connecting it from (6,inf) again we use () because we dont include the 6 in the solution so the full answer is
(-2,2)U(4,6)U(6,inf) but an easier way to type it it (-2,2)U(4,inf)-{6} which indicates that its all values from 4 to inf EXCLUDING 6 which is the asymptote that again IS NOT negative nor positive nor zero. it has no value its like a void

[midnight mirage]

wow that cleared it up so much

[celest slate]

hopefully xdd

[midnight mirage]

okay let me submit that really quickly see

[celest slate]

btw make sure that the brackets you use for the 6 are the fancy ones {}

[midnight mirage]

i dont have that option

[celest slate]

you use the fancy one when you are just speaking of certain elements and not a domain or a range

[celest slate]

you can use [6] then

[midnight mirage]

I only hae () option

[celest slate]

oh then write it as the first form

(-2,2)U(4,6)U(6,inf)

[midnight mirage]

yup worked!

wow makes so much nese bro

[celest slate]

✅

[celest slate]

thats why we love math

[midnight mirage]

ok so now i will use that to solve this problem if your free can you guide me through it?

this is the last problem

[celest slate]

sure

[celest slate]

yeah this looks fairly similar

[midnight mirage]

yeah i dont know what the dot means

[celest slate]

i will try not to give too much help unless you are really stuck

[celest slate]

so there are two types of things like that, a dot that is filled or a dot that looks like a circle

[midnight mirage]

for sure, just let me know if i am wrong if thats cool with you, i only have 1 chance of submitting it

[celest slate]

basically the filled dot is expressed with [] which means that the value is included

[celest slate]

mhm

[celest slate]

the circle one is with normal () which means that the value is not included

i could provide an example

if you want

[midnight mirage]

ok wait one sec

the shaded one is not included but the non shaded one is

[celest slate]

no its the opposite

shaded is included

[midnight mirage]

AHHH

ok so

[midnight mirage]

yes yes it makese sense now

[celest slate]

forgive me xdd thats the best i was able to provide atm

[midnight mirage]

all good!

if its included we use [

[celest slate]

yes

[midnight mirage]

which means the domain should be, [-14,-inf)

becayse it goes down after that

[celest slate]

when expressing the domain you are looking at the x values

and yes even tho the graph is going down are the x values going negative or positive?

[midnight mirage]

lemme see

yeah wait nvm they are going positive

only thing thats even close to - is -11.5 but its 0 which is neutral

so

[-14,inf)

[celest slate]

yes!

[midnight mirage]

ohhh so thats how it works

[celest slate]

not that specifically

[midnight mirage]

ah

[celest slate]

but if you look at the graph

its going to the right side right?

and in the right side the x values are getting bigger

its going like

-14

[midnight mirage]

ohh yeah mhmh

[celest slate]

-13 - 12 -11

[midnight mirage]

yes yes

[celest slate]

all the way to infinity

[midnight mirage]

and the Y values are positive

[celest slate]

thats a different matter

[midnight mirage]

ah

[celest slate]

the y values are not always positive

you can see it yourself that the y value in a certain place is positive and in the other is negative

[midnight mirage]

they are positive till 2

and after that it goes neative

[celest slate]

mhm

exactly!

[midnight mirage]

kk

ok now range

[celest slate]

range, we look at y now

[midnight mirage]

highest point is at y = 6 and lowest isnt determined so

[6,-inf)

[celest slate]

but where is it going?

perfect

[midnight mirage]

infinity and beyond

[celest slate]

yessss to negative one

[midnight mirage]

ok now x intercepts are

(-11,0),(2,0)

[celest slate]

there we go

nice

[midnight mirage]

ok so zeros are just

-11 and 2

[celest slate]

mhm

[midnight mirage]

wait kebab

1 question

[celest slate]

yea?

[midnight mirage]

you see where the graph touches at 0,2

does that count as a zero or nah?

[celest slate]

good question, now there its intercepting the y axis which means f(x)=2 and not 0

the thing is at point (0,2) x is 0 not y which means what you are doing is f(0)=2

but what the question wants is what values make f(x)=0

another way to think of it

when it asks you to find the zeros of f(x), since y=f(x) we need y to be qual to 0 so that f(x) is 0

and at that point y is not 0 its just 2

[midnight mirage]

ohh ok makes sense

now, e

-9 = 2

and f 0 = 2

[celest slate]

yeah

and f(0) is also 2

[midnight mirage]

yup!

[celest slate]

but those are not what its asking us

[midnight mirage]

oh what

[celest slate]

its asking us the zeros of f(x) so we want y to be 0

and at which points y is zero?

[midnight mirage]

when x = 2

[celest slate]

yes and?

is there anywhere else y is 0?

[midnight mirage]

-11

[celest slate]

yesss

[midnight mirage]

i think its only asking for 1 point

so we should be good

but f(-9) = 2 though right?

[celest slate]

yes it is

[celest slate]

for 1 point?

[midnight mirage]

when f(0)

[celest slate]

oh wait i was still at the zeros question lol

didnt realize

[midnight mirage]

loll

[celest slate]

what is the value of y when x is 0?

[midnight mirage]

2

[celest slate]

yes

so they are both equal

[midnight mirage]

ye

ok now

H

2

[celest slate]

true

can you determine which points?

[midnight mirage]

yup

-3,-7

[celest slate]

nicee

[midnight mirage]

ok now I

[celest slate]

yeah

going pretty solid

[midnight mirage]

ok now I

3

[celest slate]

good job

[midnight mirage]

oh hell nah j and k again 🤣

[celest slate]

ahahahaha

dw

they dont know youre well trained now

you pretty much solved all questions without my help i just confirmed it for you

you got this

[midnight mirage]

ok so

[celest slate]

hm

[midnight mirage]

f(x) is bigger than 0 when

wait im so confused at this

[celest slate]

so

whenever f(x) is referred to then remember that y=f(x)

[midnight mirage]

ohh yeah wait

when y = 6 x = -5

so thats one of the peak points

[celest slate]

not like that

[midnight mirage]

oh

[celest slate]

you are gonna use the domain to describe it so with x values but since its asking you the values where f(x) is bigger than 0 then since y is f(x) it basically is asking you where y is bigger than 0

[midnight mirage]

y is bigger than 0 in a lot of points tho

[celest slate]

yeah

still

[midnight mirage]

so how would one write this?

[midnight mirage]

[celest slate]

so

[midnight mirage]

wait so

[celest slate]

remember that 0 is not positive nor negative

[midnight mirage]

[-14,2) U (2,-inf]

[celest slate]

like this you fused the positive and the negative together

[midnight mirage]

u cant do that?

[celest slate]

no, its asking you for the values ABOVE 0

so positive

[midnight mirage]

OHHH

[celest slate]

where is y positive?

[midnight mirage]

but it only wants me to include points when X is positive. as well right?

[celest slate]

nono x doesnt matter, you need y to be positive

in (2,inf) y is negative

its going down

[midnight mirage]

yes but before that its positive

[celest slate]

i marked it with red

[midnight mirage]

yup

[celest slate]

so we need the part before it then :DDDDDD

[midnight mirage]

its positive from x = -11 to 1

its also positve at x = -14 tho

[celest slate]

YES

yessss

sooo

unite them together

[midnight mirage]

HOLLLUPPP

(-11,1) U (2,-inf]

right?

[celest slate]

so uh first of all

inf is always with()

second, no thats not quite correct

let me give you a hint

one part of the asnwer

the first part is

[-14,-11)

in this area, y is positive

[midnight mirage]

oh

ok

[celest slate]

-14 is included because its negative

meanwhile -11 is zero

thats the first part

[midnight mirage]

but how would we write the part from -11 to 1 tho?

[celest slate]

its from -11 to 2

[midnight mirage]

wait is it the same thing?

[celest slate]

but both -11 and 2 are not included

so we use ()

[midnight mirage]

yeah

[celest slate]

yeah

[midnight mirage]

so [-14 (bc its included),-11) U (-11,2)

would that work?

[celest slate]

yes

that is correct

[midnight mirage]

oh intresting

so that just shows the points

[midnight mirage]

YUP thats exactly how I visualized it

we are just explaining 2 different squences, 1 from -14 to -11 an other from -11 to 2

[celest slate]

great, now try to do the last one

same principal just negative

[celest slate]

exactly

and uniting them with union symbol U

[midnight mirage]

ok so now its asking f(x) to be smaller than 0

which means the negative ppints

[celest slate]

mhm

yup

[midnight mirage]

so for this 1 sequence i see is from 2 where it goes downwards

[celest slate]

mhm

[midnight mirage]

but there is a non included o

should this be (2,inf) then?

thats it theres no any other pattern

[celest slate]

perfect

you got it

[midnight mirage]

actually??

wtff

[celest slate]

yesss

thats it

!

[midnight mirage]

aw wait i got the range wrong :/

for the range i wrote [6,-inf)

[celest slate]

oh

[midnight mirage]

i can still fix it no biggie but

[celest slate]

it is correct tho

[midnight mirage]

where did i go wrong?

[celest slate]

wait

my visuals might be bad on pc lemme see on phone

[midnight mirage]

here ya go

[celest slate]

ah i see

sorry im a bit tired didnt notice

so its correct

but we write from smallest to largest

[midnight mirage]

OHHH

[celest slate]

so its actually (-inf,6]

[midnight mirage]

so yes yes

[celest slate]

mhm

[midnight mirage]

what about the [)?

[celest slate]

6 is still included

[midnight mirage]

it gonna be (-inf,6]

[celest slate]

inf is never included

[midnight mirage]

so that i said is good right?

[celest slate]

that is correct

[midnight mirage]

okay

[celest slate]

as if you look at thegraph

-5 gives you 6

so you can actually get this value

which means you include it

if there was a gap so a non shaded circle then we wouldnt include it

[midnight mirage]

ahh

got it!!

[celest slate]

hope this helped

🙏

[midnight mirage]

thank you sooo muchhhh!!

[celest slate]

thank you for listening 😊 ,

good luck and feel free to ask any questions you have later

[midnight mirage]

ok one thing that confuses me is which part did i go wrong in this graph it looks accurate to me

the assympotpe is at 3 and compared it to desmos looks good enough idk i think this website is tweaking

[celest slate]

what is 1/0?

[midnight mirage]

sorry?

[celest slate]

1/0

what is the value of 1/0

what is the answer for it?

[midnight mirage]

u cant divide a number by 0

[celest slate]

which means its undefines

undefined

[midnight mirage]

i think i forgot to add holes in this one

[celest slate]

now in that graph try to plug in the number -6 and 3 for x

you will end up with 0/0 and 9/0

which are undefined

so yes, you need to add the holes

[midnight mirage]

so theres one hole in -6

[celest slate]

mhm

any other?

look at the values that make the denominator equal 0

you can make the denominator equal 0 and solve for x to find the holes

in this case you will get a quadratic equation so thats 2 solutions

[midnight mirage]

so the holes that I wrote down were at, -6,-1/9

[celest slate]

uh

well

why do you think -1/9 is a hole?

[midnight mirage]

i did the math like yesterday

[celest slate]

-1/9 does not make the denominator 0

[midnight mirage]

wait those are 0 cordiantes not holes sorry

[celest slate]

ah

-6 is a hole tho

[midnight mirage]

oh

-6,0?

[celest slate]

its not -6,0 quite because when you plug in -6 you get 0/0 which is undefined

but at x= -6 there is a hole because of that

[midnight mirage]

oh

but i think ill have to inculde it here

[celest slate]

include?

[midnight mirage]

nvm nvm sorry

oh yes i got it right!

@celest slate thank you so much for today

[celest slate]

No worries :)))

Thanks to the other guy aswell

[midnight mirage]

i will

[celest slate]

Have great day

[rustic goblet]

@midnight mirage you should close this channel if you're done here

[midnight mirage]

oh yeah sorry i just got back

.close

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[west hedge]

What is the onomatopoeia for crying

[west hedge]

I’m trying to get the intersection

But I keep getting a wrong answer

[rustic goblet]

“waah” would be my first thought

[west hedge]

That’s for the babies

[west hedge]

Can someone check this for me

<@&286206848099549185>

Load Mickey, please save me.

Save me from this purgatory

Guide me to the truth

I will serve you all my life.

[magic junco]

If you cannot find a set of (t,s) that fulfills both line parameter, the two lines do not intersect

[west hedge]

Why I get a wrong answer

Why I get t=-3

Is it god’s plan

To humble me

[magic junco]

What set of (t,s) did you get?

[west hedge]

Both t and s equal -3

[west hedge]

The process

[magic junco]

Where did you get this from?

[west hedge]

The first equation and second equation imply t=s

[magic junco]

No, it doesn’t

[west hedge]

I see

Thank you lord Mickey

[magic junco]

You’re welcome

[west hedge]

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[safe radish]

@rough pelican Has your question been resolved?

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[safe radish]

What step are you on?
1. I don't know where to begin.
2. I have begun but got stuck midway.
3. I got an answer but I was told that it's wrong.
4. I got an answer and would like my work checked.
5. I have a question about someone else's work/solution.
6. I have completed the problem and don't need help anymore. Thank you.
7. None of the above

@modern bloom Has your question been resolved?

[tardy mango]

Reposted here: #calculus message

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[naive shoal]

In a shipment of 75 soccer balls there will be 4 defective ones and 71 good ones. If you buy 3 soccer balls from the store, what is the probability that you will get 2 good ones and 1 defective one?

[quick crater]

the number of ways to pick 2 good balls is to choose 2 from 71, the number of ways to pick a defective ball is to choose 1 from 4

[naive shoal]

ohh

that makes sense, thank you!

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[safe radish]

@spiral saddle Has your question been resolved?

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@spiral saddle Has your question been resolved?

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[storm ice]

What is 0^0?

[storm ice]

Because I thought that we were taught that it's well, 0.. But there seems to be quite a contradiction around it and now im not sure what to believe xd

[wise saddle]

theres not a consensus

0^0 is a convention anyway

depends how you look at it, for example in algebra it can make sense to say 0^0 = 1

but if you look at it differently, 0^0=0 may make more sense

so anyway, it depends

[storm ice]

Welp

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[leaden harness]

What is the equation of the circle passing through (-4,-4) and tangent to the line 2x-3y+9=0 at (-3,1)?

1.) What's the best way to do this?
2.) How do I do this?
3.) I'm clue less

[safe radish]

@leaden harness Has your question been resolved?

[thin bridge]

start by doing a rough sketch

[leaden harness]

alright

Tangent to the line 2x-3y + 9 = 0

[thin bridge]

it is useful to apply that the centre of the circle will be on the normal line

[leaden harness]

How do I plot this

[thin bridge]

its the equation of a straight line
determine two points
then draw a line through it

[leaden harness]

do I have to make it in the form of mx + b = y?

[thin bridge]

you can if you want

[leaden harness]

Sorry for the late reply I was eating

[leaden harness]

So I technically don't have to? How would that work?

Sorry my knowledge in lines and stuff are really bad

[thin bridge]

sub in a value for x or y, then solve for the other variable (this will get you one point)
then repeat

getting

mx + b = y
first is usually a good idea regardless

[leaden harness]

I see alright

(-4,-4) , (-3,1) either values are ok for subbing in one of the x and y values for 2x-3y + 9 = 0 right?

[safe radish]

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[kindred sentinel]

How do i solve 1 b?

how can i find a domain and range of a composite function with just the coordinates?

[weak flax]

wait hold on

shouldn't it just be (-infinity, infinity)?

[shadow verge]

No

[weak flax]

oh sorry i was looking at the graph my bad

[shadow verge]

Range won't be

The two functions are y = |x|
And y =x² -4

So Range of first function is [0,∞)
2nd one is [-4, ∞)

[jade magnet]

i dont think they are talking about the functions in the graph

[kindred sentinel]

yeah

i wasnt

but i solved it already

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[uneven tangle]

how do i solve this?

[uneven tangle]

after the -12ab is a comma btw

[twilit spindle]

you subtract them in "pairs"

in relation to common exponents

so 2a^2b^3 can be subtracted from -4a^2b^3

but not 8a^4b^3

then just do the subtraction :)

[uneven tangle]

i reall dont know where to start because i watched videos and they applied things such as distribution and vertical method

and i dont know how to apply it in this one

[lime dust]

Start by subtracting equal things

Imagine a^2b^3 are potatoes

[shadow verge]

What ?

[lime dust]

So u have -4 potatoes in the first polynomial and 2 potatoes in the second

U subtract them like this

[shadow verge]

😅

[lime dust]

Is the same concept

After you finish with equal things, we can try working the rest

But start with simple things

[uneven tangle]

if a^2 - a^2 its = 0 ?

[fast mason]

Yes

[uneven tangle]

therefore

2a^2b^3

-(-4a^2b^3

=-2?

[wispy monolith]

@uneven tangle calculate the coefficient of same polynomial like
5a²-2a²
So 5-2=3
3a² it will become

[wispy monolith]

6a²b³

[uneven tangle]

oh nvm we subtracting here

so its just the coeffecient that gets changed when i do these operations?

[wispy monolith]

Yes

[uneven tangle]

okay now i got this

[fast mason]

You can look at a^2b^3 sort of like a single variable when doing subtraction and addition

[uneven tangle]

how about the rest?

[wispy monolith]

Unless it's multiplication

Then polynomials will also change

Or division

[wispy monolith]

Subtraction addition only coefficient
Division multiplication both coefficient and polynomial

[uneven tangle]

ok now that we got it, what do i do with the rest?

since i cant apply the same thign bc they have different exponents

[wispy monolith]

You leave them as it is, signs will change for 2nd polynomials

Because you are subtracting 2nd one from first

[burnt notch]

Think ok a²b³ as 🍎

So you have 2 🍎 - (-4) 🍎 which equals 6 🍎

[uneven tangle]

yes i got it

a - (-b) = a+b

[wispy monolith]

Bingo

[uneven tangle]

with this instructions does it go like this?

(2a^2b^3 + a^4b^3 + 8ab^4 - ab)

• (-4a^2b^3 -ab^4 + 8a^4b^3 - 12ab

---

bruthj this discord

the dot indicates as a - sign

[wispy monolith]

Yeah you just need to subtract now, make sure to change the signs of 2nd one while subtracting

[uneven tangle]

correct me if im wrong

say we have (2a^2 + y^2) - (-4^a2 - y^2)

does the 2nd binomial go like - (+4^a2 + y^2)

is that how it works? or nop

[burnt notch]

<@&268886789983436800>

[stone plinth]

mb

wrong channel

im so sorry

[wispy monolith]

Suppose we have (2a^2 + y^2) - (-4^a2 - y^2)
2nd binomial go like (+4^a2 + y^2)

No need of any negative sign anymore

[stone plinth]

server i mean

[burnt notch]

Don't worry

[uneven tangle]

hey i will probably have to take my time to answe this (the problem i sent) but i will close this help channel for now since i already got what u meant. can i ping u later and check my work if its correct?

[wispy monolith]

Sure

[uneven tangle]

thank you

.close

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[uneven tangle]

.reopen

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✅

[uneven tangle]

@wispy monolith since i got what u meant by the a - ( -b ) = a + b

i also rewrote it by distributing the - sign and making them all together and combining like terms, i felt like this is way more easier but idk if im correct so please check it. the number 3 is my final answer and the order is indicated on the right side.

[wispy monolith]

Seems good bro

[uneven tangle]

HELL YEAH

thank you so much bro

[wispy monolith]

Anytime

[uneven tangle]

❤️

i hope you have a good time around, long live

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[viral grove]

wait nvm i get it i just cube root perimeter

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@viral grove Has your question been resolved?

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[stuck frost]

No idea how to solve this

Can some one help?

[mellow cypress]

What is a requirement for matrices to be summed?

[stuck frost]

To be in the same detention

So in this case 2 by 3

[mellow cypress]

Im guessing you mean to have the same dimensions

But yes

[stuck frost]

Autocorrect lol

[mellow cypress]

And what dimensions should X have such that when left multiplied with the other will give you a 2 by 3?

[stuck frost]

2 by 3

[mellow cypress]

Yeah

With that information, do you know what to do next?

[stuck frost]

Make a 2 by 3 matrix with letters from a to f and make it equal to x

[mellow cypress]

Yeah, then just multiply, sum and equate

[stuck frost]

Ok here is the final result

[mellow cypress]

And for rhs?

[stuck frost]

2x

[mellow cypress]

What did you use for X on lhs?

[stuck frost]

A 2 by 3 matrix with letters from a to f

So i replaced 2x with the 2 by 3 matrix of 2a .. 2f so what do i do now

[mellow cypress]

Equate them component wise and you should get some equations to solve simultaneously

[stuck frost]

Ohh i found it

I subtracted the 2x matrix and made it equal to a 2 by 3 0 matrix

And now i need to make 2 other matrixes with the intent to find a b c d e f

[safe radish]

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[narrow chasm]

How do I solve this

[safe radish]

@narrow chasm Has your question been resolved?

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@narrow chasm Has your question been resolved?

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@narrow chasm Has your question been resolved?

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[misty spade]

i would be extremely appreaciated to anyone who solves this problem of mine

[fast badge]

the first eqn has two roots

what are they?

find those roots

then for those values of x

plug them into the second equation

and set the value to 0

and get the potential values for a

thats it

[misty spade]

thanks

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[winter reef]

How do I factor y=-2sin(2pix)+1 between the interval [0,1]

[obsidian oracle]

what do you mean by factor

[winter reef]

substituting y for 0 and solving

[obsidian oracle]

so you mean solving for x in -2sin(2pix)+1 = 0

[winter reef]

yes

[obsidian oracle]

well

if -2sin(2pix)+1 = 0, what's the value of sin(2pix)

[winter reef]

0.5?

[obsidian oracle]

yes

[ivory obsidian]

-2sin(2pix)+1 = 0
1 = 2sin(2pix)
1/2 = sin(2pix)
2pix = 1/6 pi
x = 1/12

[obsidian oracle]

1. not the only solution. 2)

!nosols

[safe radish]

As a helper, please do not give out answers that could be copied as a homework solution. Have the student work through the problem themselves and guide them along the way.

[obsidian oracle]

since we're solving for x in [0,1], that means 2pix is in [0,2pi] right

which angles in that range have sin(angle) = 0.5?

[winter reef]

30 or pi/6

so that means the solutions for x in the interval [0,1] is 1/12 because you get 2pix = pi/6)/2pi -> x = 1/12

[winter reef]

I didn't ask for the answer man

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[obsidian oracle]

this is not the only angle

pi/6 is in the first quandrant

[winter reef]

5pi/6

[obsidian oracle]

but there's another point on the trig circle that's at the same height

[obsidian oracle]

exactly

and this angle leads you to x = ?

[winter reef]

5/12

[obsidian oracle]

alr you got it

[winter reef]

Thank you

[safe radish]

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[foggy bay]

T, S in End(V) are linear projections
Find a Necessary and sufficient condition on the Kernels and Images in order for ST to be a projection. ( (ST)^2=ST )

Can anyone please help me with this question I have been trying to solve it but failed

[safe radish]

Please don't occupy multiple help channels.

[foggy bay]

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[balmy adder]

honestly just need a little help, a starting point
ik my properties and stuff, but this just got me confused, don't even know where to start on it

[red delta]

The whole thing?

Or just a specific number

[balmy adder]

nah just #1

[cunning pasture]

what is E cap?

angle?

[balmy adder]

yep

[red delta]

sum of angles in a triangle is...

[balmy adder]

180

[red delta]

triangle ADF is isosceles right

[cunning pasture]

angle E = angle DFC

chase the angles

[safe radish]

The purpose of this server is to help you learn, not to hand out answers. Do not ask someone to give you the answer directly.

[balmy adder]

yep

2 sides =

[red delta]

And angles

The two equal angles here are  and D

[balmy adder]

so a2 and d1 =, gotta be 45 each but idk what the reason is

[red delta]

Sum of angles in a triangle is 180⁰

[red delta]

90 + 45+45=180⁰

[balmy adder]

alr bet so I don't gotta mention isosceles right?

[red delta]

You do

That's why the angles are equal

So can you find Ê

[balmy adder]

I mean like for the reason, usually we say (Int angles of triangle = 180) but I don't remember where I put the isosceles lmao

[balmy adder]

yep, got some time if I get stuck so I believe it'll be fine lol

[red delta]

You put the isosceles where you state a2=d1

Like a2=d1[isosceles triangle]

If that's all type .close

[balmy adder]

ohhh I see

thanks

.close

[safe radish]

Channel closed

Closed by @balmy adder

Use .reopen if this was a mistake.