functions

[nocturne sierra]

guys can someone explain how we do part b, its a silly q but im confused

like i subbed in 4 for x

1/5

but thats wrong

[hollow tangle]

yh because

what about 3.4

or 3.2

[nocturne sierra]

okay so what then

1/1+3.4

[hollow tangle]

take the limit of f as x approaches 3

in other words try plugging 3.2, then 3.1 and numbers that are even closer to 3 into the function

and see what happens

[nocturne sierra]

it tends to 0

gets smaller eachtime

[hollow tangle]

aight

now what u wanna do

[nocturne sierra]

idk u tell me

[hollow tangle]

is take the limit of f as x goes to inf

[nocturne sierra]

okay so how do i do that

[hollow tangle]

basically see what happens as u plug in larger numbers for x

[nocturne sierra]

wait sorry let me rephrase from earlier

if we add in big values

such as 10

itll go smaller and tend to 0

if we add smaller values

it goes further away from 0

[nocturne sierra]

tends to 0

[hollow tangle]

wait

what form of f are you taking limits of

the one from the q or the one from part a

[nocturne sierra]

one from a

[hollow tangle]

right

[hollow tangle]

this is correct

[hollow tangle]

not this

[nocturne sierra]

that’s why

i said

let me rephrase

okay but now what do i do with this q

[hollow tangle]

so for large x f goes to zero

for x approaching 3

f goes to 1/4

[nocturne sierra]

how do we know 1/4

tho

is the biggest

value

[hollow tangle]

so as x is increasing f is decreasing isnt it

[nocturne sierra]

yeah

[hollow tangle]

yeah so as x goes away from 3

f goes to zero

as x gets closer to 3

f gets larger

if that makes sense

[nocturne sierra]

yeha so 0 is one

range

but

to 1/4 idk how u attain this number

why is it 1/4

when x needs to be greater than 3

so 1/5

surely

[hollow tangle]

you can just plug in x = 3 here to find the limit

yes the function isnt defined at x = 3

but we know its going to approach 1/4 as we get closer to x = 3

ok plug in x = 5, x = 4, x = 3.5, x = 3.25 , x = 3.0000015

tell me what you get for each sub

[hollow tangle]

alright sure, you've plugged in x = 4 here but f is defined for any real number greater than 3

so what about 3.5 or 3.2 or 3.0000000000000000000000000000000000000000001

[nocturne sierra]

x=5 -> 0.1667 , x=4 ->0.2 , x=3.5 ->0.222 ., x=3.25 -->0.23 x=3.0000015->0.2499999

[hollow tangle]

what do you notice

as x gets closer to 3

[nocturne sierra]

leans to 0.25

[hollow tangle]

exactly

you see why its 1/4

[nocturne sierra]

wait it makes sense

so itll never touch 0

[hollow tangle]

this is the whole idea behind limits

[nocturne sierra]

do u think u can give me one to try?

[hollow tangle]

nah

but it'll get infinitely close

might help to graph it and see for urself

[nocturne sierra]

it makes sense

thank you

[hollow tangle]

u want a limits question

?

[nocturne sierra]

another eg of what we just went through yeah

[hollow tangle]

let me think of something

err

[nocturne sierra]

its okay

i asked

chatgpt

for one

[hollow tangle]

f(x) = x+1/(x+1)^2

domain x > -2

[nocturne sierra]

ohh thats a harder one

ok wait

first step

do i sub in -2

[hollow tangle]

ye

[nocturne sierra]

so

0,-1

r the limits

[hollow tangle]

this is a trick question btw

[nocturne sierra]

why u tricking me

bit devious

so 0 is wrong

theres no limits w this one

i dont understand

cant be -2 bc its undefined

[hollow tangle]

mb i had to go

i had another goose being dopey moment

but the range is

f(x) >= 0

or

f(x) < -1

so you did get it you just had to write it as an inequality

well done

the trick part

was in the domain, i wanted to see if youd spot the domain is wrong

[nocturne sierra]

wait why is it bigger than -1 and or equal to 0?

should it not be bigger than 0

i dont understand

[hollow tangle]

it is

[hollow tangle]

see

there's two parts to the range instead of one like in ur question

[nocturne sierra]

wym

yeah

i know

but

why is it equal to 0

should it not be >0

[hollow tangle]

oh yh i put >= 0

my mistake

ur right

[nocturne sierra]

what about the -1?

[hollow tangle]

what about it

why its < -1 ?

[nocturne sierra]

yeah

[hollow tangle]

f isnt defined at x = -2 remember

[nocturne sierra]

hang on im so lost now

yeah it isnt definef at -2

but what has it got to do with -1

[hollow tangle]

what does the range tell you

[nocturne sierra]

bigger than -2

x>-2

so we can start at like -1.999999

[hollow tangle]

i meant what does the range of a function tell you

but anyway it tells you what all the y values are

so if u sub in x = -2 into f you'd get -1

since f isnt defined at x = 2

then you know that for values around x = 2 you get values around -1

u get it

[nocturne sierra]

okay i understand , so why the inequality <-1

[hollow tangle]

sub some values in that are either side of -2

and see what u notice

hint: ||might help to notice that you can simplify f into 1/(x+1) where x != -1||

[nocturne sierra]

i already did simplify lol

ikm just confused tbh

we have the equation 1/(x+1)

[hollow tangle]

function

but yes

[nocturne sierra]

man idk why its bugging me so much

mb

idk

im so bad at this

[hollow tangle]

lmk what ur confused about

[hollow tangle]

did you do this

[nocturne sierra]

idk what u mean tbh

[hollow tangle]

-2

the number -2

sub in values which are either side of -2

so for example -3 and -2.5

-1.8 and -1

have you graphed this

it will help

[nocturne sierra]

ibr idk how to graph it

im genuinely so bad

like how would i graph this idk

all i know

is that

it will look like this

[hollow tangle]

yeah u got it

[nocturne sierra]

yeah but no the values tho

[hollow tangle]

almost

1/x+1 is just the graph of 1/x

but shifted one unit left

[nocturne sierra]

ohhh

yeah thanks for reminding

okay

so

now what tho

bc im still not understanding thi s -1 business

[hollow tangle]

shit sketch but hopefully u get the idea

[nocturne sierra]

this is stressing me out

yeah

i do

[hollow tangle]

right so

if we sub in x = -2 into 1/x+1

we get -1

correct

[nocturne sierra]

yeah we do

[hollow tangle]

alr

so as x gets further away from -2 you can see that the graph shoots downwards (before it starts shooting upwards again)

its just gonna get smaller and smaller

and go off to negative infinity

so our range of values are from -1 all the way off to negative infinity

not including -1 btw

[nocturne sierra]

why not including -1

[hollow tangle]

which is why we have f(x) < -1

-1 is the output from inputting -2

if f is not defined at -2

how can you input it

[nocturne sierra]

ohh so because were getting -1 when we input -2 it is baso not defined

[hollow tangle]

if you cant input -2 into the function then you cant get an output

which would be -1 here IF the function was defined at x = -2

[nocturne sierra]

yep

so we ignore -1

and say

we <-1

ahhh mkaes sense

makes

[hollow tangle]

perfect

[nocturne sierra]

omg idk why that took me so long to understand

thank you

[hollow tangle]

these domain and range qs throw a lot of people off

[nocturne sierra]

yh ngl im not stuck on another one💀

[hollow tangle]

😂

[nocturne sierra]

if we input 1/2

we get undefined

[hollow tangle]

yh

look at the domain

thats why its >

[nocturne sierra]

so what would be the answer for this

[hollow tangle]

part c is it

[nocturne sierra]

yeah

[hollow tangle]

its only 1 mark

because of the relationship between domains + ranges of a function and the domains + ranges of its inverse

how is the domain of f linked to the range of f^-1

[nocturne sierra]

hang on not part c

wait yeah part c

omds idek anymore man

i clearly dont understand this topic

[hollow tangle]

u should know that the domain of a function f

is the range of its inverse f^-1

and the range of f is the domain of f^-1

[nocturne sierra]

yeah ik its vice versa lol

[hollow tangle]

yh so its just that

[nocturne sierra]

so its x>1/2?

[hollow tangle]

no

[hollow tangle]

read this again

[nocturne sierra]

so the domain is x e R

ngl man

i clearly dk

anyhting

[hollow tangle]

the domain is x E R

and x > 1/2

do u understand what this means

[nocturne sierra]

no

[hollow tangle]

youve been doing this for a while

take a break

and come back

[nocturne sierra]

i havent tho

[hollow tangle]

the E thing means 'element of' and R means 'the set of all real numbers'

[nocturne sierra]

how have i been stuck on 1

q

for an hour

this is ridiculous

[hollow tangle]

ts happens to me all the time

honestly ur fine

[nocturne sierra]

nah im rlly angry inside

ibr

[nocturne sierra]

yeahh i know this

so what is the domain

and what is the range

[hollow tangle]

you want the domain of f^-1

and you know that its the same as the range of f

so what do you need to find

not a trick question this time 😭

[nocturne sierra]

what i understand is

lets say we have f(x) domain and range

for f(x)^-1 the domain is the range

and the range is the domain

theyre swapped right

[hollow tangle]

ye

[nocturne sierra]

so what is the domain in this q

x e R

right

[hollow tangle]

both x E R and x > -1/2

[nocturne sierra]

okay so whats the range

[hollow tangle]

simply means x is a real number which is greater than 1/2

thats what u gotta work out

[nocturne sierra]

bruh

idk how to work that out

[hollow tangle]

no you can

same process as the last q

[nocturne sierra]

okay well do we sub

in

-1/2

i mean

1/2

we get an undefined

[hollow tangle]

yes

dyk what this actually means tho

visually

[nocturne sierra]

no

[hollow tangle]

ok think of y = 1/x

the graph

[nocturne sierra]

yeah

[hollow tangle]

undefined when x is 0

whats happening

[nocturne sierra]

it doesnt touch 0

[hollow tangle]

yes

and it shoots upwards

it goes off to infinity

this is why its said to be undefined

[nocturne sierra]

okay so now wha

what

so it has to be bigger

than 0

[hollow tangle]

so with 1/2x-1 the same thing happens when plugging in 1/2

what you wanna do now is consider large values of x

[nocturne sierra]

such as

?

[hollow tangle]

upto you

[nocturne sierra]

okay lets go with 2

[hollow tangle]

sure

they can be as wild as u want

[nocturne sierra]

im not feeling that wild

[hollow tangle]

lmao

alr then so what do you get

[nocturne sierra]

0.33333

[hollow tangle]

ok

now try x = 5

then try x = 10

and then try x = 32479874498

[nocturne sierra]

0.11111

[nocturne sierra]

0.0526

[nocturne sierra]

1.539x10^-11

[hollow tangle]

thats a very tiny number

basically zero isnt it

[nocturne sierra]

yeah

[hollow tangle]

so u see the trend here?

whats happening to the outputs as x gets larger

[nocturne sierra]

going towards 0

[hollow tangle]

yes perfect

whats the range gonna be

[nocturne sierra]

how would i know

theres too many numbers

to input

[hollow tangle]

you dont need to input every single number

you already know where the function is heading

as x deviates from 1/2 and gets larger and larger f goes to zero

and as x gets closer and closer to 1/2

[nocturne sierra]

so does it have to be bigger than 1/2

[hollow tangle]

f goes to infinity

so the range is f(x) < 0

[nocturne sierra]

why is it smaller than 0

[hollow tangle]

sorry > 0

[nocturne sierra]

why am i strugglign with something like this 😭

[hollow tangle]

look yh

just remember two questions when finding the range

what happens as x gets stupidly large or stupidly small

what happens as x gets close to x = k

in this case k is 1/2

as the domain was x > 1/2

anyway now u got the range you have ur answer for part c

[nocturne sierra]

unfourtanetly this is going to be one of the silliest topics i get stuck on

i still dont understand it well💀

i have a headache man

[hollow tangle]

what part dont you get

[nocturne sierra]

it just doesnt make sense overall

i think i need a tutor for this

[hollow tangle]

do you know why im doing what im doing

[nocturne sierra]

um

only 5%b of it

[hollow tangle]

right

what im doing is looking at the extremes

if my domain is x > 5

then im only interested in how the function behaves at values that are super close to 5

and at values that are very large

i could look at x = 6,7,8,9 but thats not really gonna help much

but if i look at x = 10 billion for instance

[feral tulip]

Why not 10 gazillion

[hollow tangle]

then im able to see how the function is behaving

as x just flies off to infinity

[feral tulip]

How do you know it's not walking

[nocturne sierra]

ngl wifibad can u stop

respectfully

ur comment r rlly irrelevant here

[hollow tangle]

what about 10 googol

[nocturne sierra]

okay so ur studying the values that r bigger than x

to see how they behave

as they tend to 0

[hollow tangle]

almost

[hollow tangle]

yes

[hollow tangle]

yes

[hollow tangle]

not necessarily

ur interested in ridiculously large numbers because they give you information which actually helps

what is that information

how the function is behaving at these large numbers

[nocturne sierra]

wtf 2 he’s

hrs

on this

my head hurts

i don’t understand

anyrbing

i think

im gonna go do smth else

i’ll come back to this

@hollow tangle thank you for being patient w me

[hollow tangle]

ofc man no worries but damn has it really been 2 hours