but, you square the number first, which will turn it positive
#precalculus
pale bison
willow bear
@pale bison no
wraith idol
trim fable
do u get what
willow bear
what you wrote on the very left is $x^{\sqrt{2}}$
trim fable
im thinking
obsidian monolithBOT
Ann:
willow bear
if $x \geq 0$, then $\sqrt{x^2} = x$.
if $x < 0$, then $\sqrt{x^2} = -x$.
obsidian monolithBOT
Ann:
willow bear
@trim fable
willow bear
i wrote everything.
willow bear
and?
willow bear
if $x \geq 0$, then $\sqrt{x^2} = x$.
if $x < 0$, then $\sqrt{x^2} = -x$.
this requires no further comments.
trim fable
every number would be
obsidian monolithBOT
Ann:
willow bear
$\sqrt{x^2} = |x|$ literally by definition.
obsidian monolithBOT
Ann:
willow bear
do you understand this or not.
willow bear
no
willow bear
i literally just explained it
trim fable
ok
trim fable
ok..
willow bear
if $x \geq 0$, then $\sqrt{x^2} = x$.
obsidian monolithBOT
Ann:
willow bear
do you understand this or not
willow bear
is it true that $-5 \geq 0$?
trim fable
its not (-5)^2?
obsidian monolithBOT
Ann:
trim fable
no
willow bear
no, it is not true that $-5 \geq 0$.
obsidian monolithBOT
Ann:
trim fable
lol
willow bear
if $x < 0$, then $\sqrt{x^2} = -x$.
obsidian monolithBOT
Ann:
willow bear
do you understand this or not
trim fable
yes?
but u wont answer my question
u wont square the negative with the number
just the number?
willow bear
your question makes no sense
trim fable
so its technically - (5)^2 = -25 not (-5)^2=25?
willow bear
if the number is -5 for example
its not (-5)^2?
what's it?
what's "it"?
what's "it"?
what's "it"?
trim fable
the answer
willow bear
what answer
willow bear
YOU'RE NOT MAKING ANY SENSE.
trim fable
of rip
willow bear
WHAT EXPRESSION.
uncut mulch
x^2 when x= -5 apparently
willow bear
IF $x = -5$, THEN $x^2 = (-5)^2$.
obsidian monolithBOT
Ann:
trim fable
but what i was saying was
willow bear
no stop
trim fable
ok
willow bear
there are exactly two real numbers which when squared give x^2
x and -x
do you understand this or not
trim fable
oh ok
willow bear
for any given value of x, sqrt(x^2) will by definition have to be one of these values.
it will be precisely the one of these values that is positive.
do you understand this or not.
trim fable
yes
willow bear
okay.
trim fable
what im saying is it will ALWAYS BE + ?
willow bear
I'M NOT DONE YET.
willow bear
when x is positive, -x is negative, and therefore sqrt(x^2) = x.
do you understand this or not.
willow bear
when x is negative, -x is positive, and therefore sqrt(x^2) = -x.
willow bear
when x is negative, -x is positive.
trim fable
well what i was questioning actually
willow bear
do you disagree with this?
trim fable
(-x)^2 is positive
willow bear
when x is negative, -x is positive. do you disagree with this?
i am expecting one of two answers:
"yes, i disagree with this" or "no, i agree with this"
trim fable
no
?
i agree?
u mean the final
answer
when u say -x right?
like after u square?
willow bear
for any given value of x, sqrt(x^2) will by definition have to be one of x and -x.
it will be precisely the one of these values that is positive.
when x is negative, -x is positive.
therefore, when x is negative, sqrt(x^2) = -x.
willow bear
what part of what i just wrote is confusing?
willow bear
i was under the impression that the entire argument was laid out crystal clear
willow bear
and that i "meant" exactly what i said
undone pawn
lmao
willow bear
what the fuck do you have to say now
uncut mulch
no
willow bear
no
uncut mulch
you aren't squaring x there
willow bear
if x = -5, then x^2 is NOT -5^2
undone pawn
(-5)^2
willow bear
because ORDER OF OPERATIONS is a thing no matter how much you wish to forget about it
wraith idol
willow bear
what you were saying made very little sense if indeed it made any sense at all.
willow bear
no
trim fable
were u
willow bear
i did not
trim fable
talking about
The final answer????
ohhhhhh ripppppppppp
omggggggggg
what was i thinkinggggggggggg
i thought u meant
undone pawn
bruh
willow bear
if by "the final answer" you mean "the value of sqrt(x^2)"
undone pawn
that was basically expanding the |x|
trim fable
OMGG
willow bear
THEN YES OF FUCKING COURSE THAT IS WHAT I WAS TALKING ABOUT
undone pawn
wot m8
grizzled orchid
trim fable
coz ur squaring itttt
undone pawn
what the fuck are you saying
willow bear
what
trim fable
so i started questioning u
grizzled orchid
hi ann
uncut mulch
whut?
grizzled orchid
are we in hell again today
uncut mulch
the x is clearly negative in the second case as stated by "x < 0"
trim fable
its absolute wow
undone pawn
you mean the square root?
willow bear
no LOOK
undone pawn
yes that is always positive
trim fable
YES
willow bear
JUST BECAUSE A MINUS SIGN APPEARS IN FRONT OF A VARIABLE
willow bear
DOESN'T MEAN THE VARIABLE ITSELF "IS NEGATIVE"
trim fable
i thought he meant after squaring u got a - coz i thought he was doing
undone pawn
if x is positive, then +x is positive
willow bear
who's he
willow bear
i'm a she
willow bear
k
undone pawn
aight
wraith idol
Damn
undone pawn
okaaay
willow bear
undone pawn
no u
trim fable
no u
fluid shore
Why?
trim fable
u were right but i was too
fluid shore
You have nothing to apologize for
willow bear
hhh
fluid shore
Just learn as much as possible and move on.
willow bear
i feel like i lost a few thousand think pan cells from this
trim fable
oh no..
IM SORRY
gives u my brain cells
tho idk if
u want them
they are pretty useless it wont
make a difference
lol
willow bear
it's ok you can keep them
fluid shore
Ann, you worked yourself up over something like this.
fluid shore
Anyways, Star, just try to understand what Flynn and Ann were telling you before
uncut mulch
are there still parts you aren't sure about?
trim fable
i just thought they meant something else
undone pawn
ohh
trim fable
tho the final answer in the end was
agreeable
i guess
ye so
are u all mad at me ;-;
fluid shore
It's okay. There was no reason for them to work themselves up like that.
trim fable
sorry guys
fluid shore
There's no reason to be mad at you.
fluid shore
Like what ramonov asked, is there anything you're unsure about?
slim lintel
jesus christ
fluid shore
Alright.
fluid shore
There's no reason for ANYONE here to get irritated just because you're asking questions.
trim fable
there u got the english u wanted @fluid shore
uncut mulch
just don't use freshman's dream again
trim fable
what do u mean @uncut mulch
this is math related
math related emotions
EVERYONE
there lol
uncut mulch
earlier you implied that the sqrt was linear
trim fable
ME?
uncut mulch
yes
trim fable
oh rip when
uncut mulch
sqrt(x^2 - 2) โ x - sqrt(2)
fluid shore
Idk lol, it's a term that some of them use for some incorrect stuff
Anywyas, if you're stuck on something, just ask questions.
willow bear
in general yes this is false
trim fable
ok
willow bear
freshman's dream is sth different
trim fable
sth?
uncut mulch
something
trim fable
oh
fluid shore
,w graph y = x
pale bison
how many freshman's dream are there,
are there an undergrad's dream?
obsidian monolithBOT
Requested by Abhijeet Vats#7807
willow bear
freshman's dream is $(x+y)^p \overset{?!}= x^p + y^p$
fluid shore
,w graph y = abs x
trim fable
huh
fluid shore
Just wanted to give you a visual image of it.
trim fable
yes
willow bear
freshman's dream is $(x+y)^p \overset{?!}= x^p + y^p$, for $p \neq 1$. most often encountered with $p = 2, 3, \frac12, -1$.
trim fable
what
fluid shore
Well, that's usually a good thing to do but making mistakes is absolutely fine.
obsidian monolithBOT
Ann:
trim fable
im in grade 12 and im making dumb mistakes like that
thats honestly pretty sad
tho i am good at it i just do random stuff at timed
fluid shore
Well, then, try not to make such mistakes in the future
fluid shore
If you want to learn this stuff in greater detail, i'd suggest using something like brilliant.org
trim fable
ok
fluid shore
Use the wikis there to learn the material and solve problems there as well.
trim fable
ok
undone pawn
abhijeet is the brand ambassador for brilliant
trim fable
๐ฎ
fluid shore
Well, I think it's a fantastic resource.
pale bison
did someone mentioned brillian.org, our sponsor for todays video
undone pawn
^
pale bison
honestly, i'm like iffy on what brillian's doing in terms of building their own wiki page
willow bear
an entire class of fuckups
pale bison
just doesn't seem right to copy off wikipidia
undone pawn
ann how could you interrupt the part where we thank our sponsors
fluid shore
Oh they don't, from what i've used anyways. The wikis are made mostly by olympiad students who've learned the material from other sources
pale bison
use code im total sellout to get 10% off your first month of purchase
fluid shore
Now, it has branched away from olympiad material
uncut mulch
if they're sponsoring us, they're not doing a very good job
native sequoia
just in case Wikipedia goes down, but that will never happen because you have donated 
undone pawn
true
lmao
well we could technically download all of wikipedia onto an external drive at this point
fluid shore
I've contributed to their physics wikis before but that was at a time when my understanding of physics was very limited.
pale bison
there's a chinese wiki called ็พๅบฆ็พ็ง
fluid shore
I just don't understand why this devolved into such a massive fight. I don't even understand how it got linked in chill.
But whatever
pale bison
what fight?
undone pawn
what fight
willow bear
baidu wiki?
fluid shore
pale bison
well yeah i merely mentioned it
willow bear
baidu-pedia i guess
pale bison
i don't see a problem
willow bear
็พ is like, the hanzi for ๐ฏ or sth
pale bison
i'm suggesting that in the very unlikely of wikipidia going down, there's a chinese one
fluid shore
20 years living in singapore and i still can't speak chinese or even read it
pale bison
it means one hundred
fluid shore
I've heard it's a pain in the ass to learn
native sequoia
hundred
pale bison
i agree
willow bear
pale bison
TIL
native sequoia
I thought it was 100 horizontal lines
pale bison
how do you explain that ็ฝ means "white" then?
pale bison
lmao
viscid thistle
what
pale bison
็พ means 100
็ฝ means white
pale bison
chill... i'm just responding to his message
vernal rapids
$\dv{็ฝ} $
obsidian monolithBOT
DarK:
viscid thistle
้จ is the best looking character imho
native sequoia
็จ
sour quiver
่ตข
viscid thistle
ใๆฏใฏๅคชใฃใฆใใพใ
fast tinsel
Could anyone explain how to solve so the variables become the above?
I basically put them both on the same fraction, multiplying A with xยฒ+x+1 and Bx+c with x-1
fast tinsel
Then I take out the X's, everything in the brackets of Xยฒ(...) should be 0
same for X(...)
and the rest is equal to 1
But that doesn't work at all
fluid shore
In other words, we compare coefficients and get A, B & C
willow bear
But that doesn't work at all
wdym
you're on the right track so far, based on what you've said
what exactly does not work
fast tinsel
-A+B+C = 0
willow bear
,w expand a(x^2 + x + 1) + (bx+c)(x-1)
fast tinsel
oh
obsidian monolithBOT
Requested by Ann#0413
willow bear
$ a(x^2 + x + 1) + (bx+c)(x-1) = ax^2 + ax + a + bx^2 - bx + cx - c$
obsidian monolithBOT
Ann:
willow bear
$= (a+b)x^2 + (a-b+c)x + (a - c)$
obsidian monolithBOT
Ann:
fast tinsel
General question out of curiosity:
If I take the derivative of the integral of some function f(x), then I would get f(x), right?
But if this integral is definite, with x going from a to b, is the derivative of the integral of f(x) equal to f(b)-f(a)?
viscid thistle
it would be easier for you to check this yourself than to ask
but also think about what do you really mean when you as kthat question
is it coherent?
fast tinsel
I'm not sure what you mean by that, English isn't my first language
frozen needle
Well first, functions that look like
$$x\mapsto\int_a^xf(t)\dd t$$
aren't always differentiable everywhere, so even your first question should be answered negatively I suppose
obsidian monolithBOT
Tuong:
fast tinsel
Oh yeah, true
Perhaps in the example of the derivative of the definite integral of (cos(xยฒ)) with x going from a->b
frozen needle
you can even have differentiable functions that look like
$$x\mapsto\int_a^xf(t)\dd t$$
but which derivative isn't equal to $f$
obsidian monolithBOT
Tuong:
frozen needle
for your second question, you're asking about the derivative of the number $\int_a^bf(t)\dd t$
\ \
if you identify the number with the constant function, then its derivative is $0$
obsidian monolithBOT
Tuong:
fast tinsel
Right, that makes sense
If I could ask one more thing, say we introduce a second variable, like so:
d(integral a->x of cos(tยฒ)dt)/dx
If that's readable
What would this give us?
I can't really calculate the integral first, that's my biggest issue
Thanks for the clear explanation by the way
frozen needle
there's a theorem
basically with continuous functions everything works fine
if $f$ is continuous then $F:x\mapsto\int_a^xf$ is differentiable and $F'=f$
obsidian monolithBOT
Tuong:
fast tinsel
Oh right, that would work with what you said earlier (my second question)
Perfect, very much appreciated
frozen needle
You're welcome
valid violet
@odd helm I dont see any equation
"50 dollars per piece" means you need to multiply the number of pieces by 50
rigid sun
i think he's trying to describe a profit function
if the starting price is 50, and you are projected to sell 4,500 units, then you're equation looks right
im no sure what they are doing
odd helm
It is 50 dollar per piece= 4500 items sold
Every dollar increase means 5 less items sold
odd helm
Oh
rigid sun
im not sure what they are doing tho
im really lost on what they are doing
oh i got it
they are doing it by actual price
not increase like you did
yeah same thing
odd helm
Alright
hallow flicker
Whatโs the difference
And whatโs the use of a dx in the end of a integral
Ok so if I solve both those equestions whatโs the answer for both
uncut mulch
how much integration have you done?
hallow flicker
Like none
uncut mulch
then the answers we give will probs be meaningless to you
hallow flicker
Is it possible to answer the second one
hallow flicker
So whatโs the answer
uncut mulch
how much calculus have you done?
hallow flicker
Fuck off
Ok this is true right?
Well I have to add the d theta in the end
But this means that the d whatever is the same as taking the derivative
Yo someone answer
undone pawn
$\int \sin\theta \ d\sin\theta = \frac{\sin^2\theta}{2} + c$
obsidian monolithBOT
FlynnXD:
undone pawn
$\int \sin\theta \ d\theta = -\cos\theta + c$
obsidian monolithBOT
FlynnXD:
undone pawn
$dy = \bigg(\frac{dy} {dx}\bigg) dx $
obsidian monolithBOT
FlynnXD:
uncut mulch
that's not correct though
undone pawn
I think there's a - missing too
uncut mulch
^
uncut mulch
because the integrand isn't in terms of the thing you're integrating with respect to,
and you'll need to do some maniuplation
uncut mulch
i mean you could try, but you'll still be left with dx
undone pawn
it just depends on whether it simplifies your problem or not
in case of sinx d(sinx) it actually is easier to evaluate
uncut mulch
i reworded what i posted earlier
undone pawn
you can play around with the d(..) in many ways
that's basically the whole u-sub thing
uncut mulch
the integrand here x^2 is in terms of x so manipulation of the differential isn't needed
hallow flicker
Cool that was helpful
winter isle
@hallow flicker you're doing integrals in precalc?
viscid thistle
gnarly scope creep
hallow flicker
Iโm doing ur mom dude
fluid shore
Rekt
viscid thistle
$\mathbb{R} e^{k} \dagger$
obsidian monolithBOT
A. S.:
native sequoia
OOOOOOOOOOOOOOO
pale bison
ur mom is applicable in nearly every situation
fluid shore
Oh shit rekt
radiant night
Can anyone help me with this?
uncut mulch
do you know your unit circle?
willow bear
what's giving you trouble here? @radiant night
radiant night
Well I know how to use the unit circle, but one wasnt provided since its sub work.
And I need the theta= aswell, since it's been a while since I've done that,
willow bear
Well I know how to use the unit circle, but one wasnt provided since its sub work.
you're supposed to be able to reproduce it from scratch
radiant night
Well, we've only learned it a couple times, and I dont have the greatest memory.
viscid thistle
memorize it, or memorize the special right triangles
uncut mulch
^ do you at least understand the first quadrant
viscid thistle
you only need to memorize the first quadrant
the rest is the same but just use sin cos rules in different quadrants
A S T C
regal nest
I dont have the greatest memory
You have google for that
languid crane
||luckily we have a formula book for all formulas in hs math||
willow bear
i'd say don't memorize but i don't have enough energy rn
neat gate
The basic trig values have easy to remember patterns. (root 1)/2, (root 2)/2 and (root 3)/2 for (co)-sines. And (root 3)^1 /3, (root 3)^2 /3 and (root 3)^3 /3 for tan.
rigid beacon
Well you only need to memorize first quadrant
And then if you can draw the triangles on the axis
You're good
Because the unit circle lies on the xy plane
So
If the triangle is in the 2nd quadrant
You know that the cosine (which lies is the x-axis) is negative because in the 2nd quadrant x is negative
Cause it goes to the left, and in the plane we define left of zero as negative
fluid shore
What do you think?
trim fable
1/x^2-1
fluid shore
No, thatโs not the answer to my question.
trim fable
oh
fluid shore
My question was: what do you think is the approach to be taken?
trim fable
oh
so take 1/x^2-1 and plug in f into x
which gives u 1/(1/x)^2-1
and then 1/1/x^2-1
fluid shore
Since itโs g(f(x)), shouldnโt you be plugging in f(x) into g(x)?
trim fable
thats what i did?
fluid shore
What you did was f(g(x))
fluid shore
Oh wait nvm, i misinterpreted your working
trim fable
oh sorry
obsidian monolithBOT
Ann:
fluid shore
Yea yea lmao
trim fable
oh oops
fluid shore
Use latex to type this stuff out, star
fluid shore
$\frac{1}{(\frac{1}{x})^2 - 1}$
obsidian monolithBOT
Abhijeet Vats:
fluid shore
Thatโs what you were going for?
trim fable
yeah
fluid shore
Aight, then simplify
obsidian monolithBOT
A brief description and guide on how to use me was sent to your DMs! Please use ,list to see a list of all my commands, and ,help cmd to get detailed help on a command!
fluid shore
Uhhhhhhhhhhh
c ii looks wrong
Itโs late here and iโm tired but it just seems wrong, given what you had earlier
willow bear
yes you did
willow bear
yes c i is right
fluid shore
$\frac{1}{(\frac{1}{x})^2 - 1}$
obsidian monolithBOT
Abhijeet Vats:
fluid shore
Simplify this properly
trim fable
ok
languid crane
The feeling when u think ci and cii are related and some higher math
Tired, perhaps
viscid thistle
Can I ask my questions
serene heath
dull ice
Zoom in lmao
viscid thistle
I can provide some insights on question 6
viscid thistle
For question 6, consider the periodicity of the trig functions.
And compare it with the average rate of change
Sin, cos and tan will have an average rate of change equal to zero
Now consider the inverse trig functions
arcsin and arccos are not defined when given an input 2pi so discard those
arctan(2pi) - arctan(0) = some number greater than zero (calculator)
So discard arctan
โ
Now youโll need to test the hyperbolic function and their inverse
Which Iโm too lazy to do
gusty halo
obsidian monolithBOT
silk crystal
How do you make a rational function with an extraneous solution?
harsh cipher
Hi
is that a typo in the second line in the answer section
question says csc > theta
willow bear
no
harsh cipher
then sec theta is positive?
willow bear
the question says csc(ฮธ) > 0 not "csc > ฮธ"
harsh cipher
yes i meant that
but in the answer section is that sec ฮธ is positive?
that's a typo i think...
willow bear
anyway yeah it's a typo and they meant csc and not sec
willow bear
it's been 4 days already lmao
harsh cipher
hahahaha
vale basin
I didnt know where to ask ๐
languid crane
Explain
vale basin
We need to solve log 9 approximately
languid crane
?!?
vale basin
Yes base 10 log
vale basin
Nvm i got it
ebon raptor
hey guys! i have a math problem i have been stumped on for an hour
fluid shore
ebon raptor
so it shows me the answer is -7/-3
but i cannot get to that answer no matter how hard i try lol
ill send my "work" in a second
does it show how to get the steps
pale bison
pearson socks?
timber plinth
,w vertex of 3x^2+8x+3
obsidian monolithBOT
Requested by Darkrifts#3866
timber plinth
so, have you ever done finding the vertex of a parabola before?
pale bison
show us your step by step work
timber plinth
^
ebon raptor
timber plinth
that 16/3 is negative
fluid shore
You should really leave a blank line between each statement in your working.
ebon raptor
ah i would subtract the 16/3 from 3
fluid shore
^What you do or don't isn't the point. It's neat to do it that way.
timber plinth
then again I write stuff like $\forall a \exists b \phi_a(b)\implies \psi(c)\vdash \forall a \eta(a)$
obsidian monolithBOT
Darkrifts:
timber plinth
also yes goon
(note: ignore the thing I wrote above, it's a meaningless example of why you shouldn't write like me)
timber plinth
that looks a lot like the right answer
ebon raptor
haha awesome i was improperly messing with the faction and whole number
thank you! glad i found this discord
viral imp
$y=2\arctan{\frac{x+1}{3}}$
obsidian monolithBOT
Umma.Gumma:
viral imp
$x=3\tan{\frac{y}{2}}-1$
obsidian monolithBOT
Umma.Gumma:
viral imp
I had to find the inverse of the first function, which is the second one, no problem about that. The thing is I have to determine the domain of the inverse function as well
given that there's a tangent I would be tempted to say that all values of y are valid except -pi/2 and pi/2
however it seems that the correct domain is $y\neq\pi+2k\pi$
obsidian monolithBOT
Umma.Gumma:
viral imp
can anyone explain this?
last compass
Well when does y/2 take a value that is a integer multiple of pi/2
willow bear
an ODD integer multiple of pi/2
desert socket
Iโm having some sort of brain fart. How is (1+x^2)^1/2 * (1+x^2) = (1+x^2)^3/2. Where is the ^2 exponent thatโs being added coming from.
Or is it (1+x^2)^2+ ^1/2.
That might be it. Nvm.
rigid beacon
@desert socket (1+x^2) = (1+x^2)^1
and then it's exponent rules
a^b * a^c = a^(b+c)
desert socket
I got it. Thanks!
ebon raptor
lol i have someone making me second guess myself
the answer to that should be 3 right
proud jetty
yea
ebon raptor
ok thats what i was thinking lol caught it on a peer review and had another friend explaining some crazy shit making me second guess myself
is that the whole question that u posted @proud jetty
proud jetty
yes
unit is permutations and combinations. every letter have a different answer, u gotta solve for it.(not a multiple choice question)
slate tree
how do i do "how many code words of any length can be spelled out using gmae timles of five different litters ( including single lettered words)
also how do you do
expansioin of polynomials byeond binomial
can anyone help me please
willow bear
uh
undone pawn
wtf is gmae timles
willow bear
^
slate tree
yeah
undone pawn
well I assume the "words" don't have to make sense
slate tree
yeah i dont think so
undone pawn
just random permutations of the letters
what's your difficulty then
minimum length is 1, max is 5, take each case and figure the number of possible words
slate tree
oh i did that but looking back i realized what i did wrong
i used the wrong amount
coluld you also help explain how i do expansion of a polynomial that is greator than a bionomial
undone pawn
what does that even mean
slate tree
so binoimal expansion is when its like this (x+1) (y+1)
but im talking about something like a tri nomial
so with three of those
slate tree
yeah but how would i distribute through with a short cut method
undone pawn
slate tree
thanks
undone pawn
not sure if that's what you're asking for, but check it out anyways
we derived this in hs, cool stuff
slate tree
cool thanks
i think i also realized that for my school we used something called lines and stars method
its k9inda weird but it works
undone pawn
yeah I remember that one
valid violet
@viscid thistle 
pale bison
it's g o o d p r a c t i c e tho
willow bear
paku dropped a question here and went offline presumably to wait until the answer is handed to them
karmic wave
hopefully they went to find a book that explains how parentheses work
summer monolith
This is a misprint too, right?
pale bison
yes
summer monolith
3 misprints on one page. They've gotta be shitting right into my soul.
languid radish
For computers, what base is the log function set to by default?
willow bear
wdym by "for computers"
severe prawn
2
willow bear
uh
no
first off it's likely not 2
and second you really have to ask your programming language's documentation
not us
severe prawn
:d
sturdy haven
fluid shore
Well, suppose $log_{10} x = y$
obsidian monolithBOT
Abhijeet Vats:
fluid shore
Then, you agree that:
$x = 10^y$
obsidian monolithBOT
Abhijeet Vats:
fluid shore
?
fluid shore
No. $x = 10^y$, correct?
obsidian monolithBOT
Abhijeet Vats:
trim fable
oh
fluid shore
What did we define y as, initially?
fluid shore
Nope
fluid shore
$a^x = y \iff x = \log_{a} y$
obsidian monolithBOT
Abhijeet Vats:
fluid shore
That's how you get logarithms
trim fable
yeah im not good at logsss ahhhh
fluid shore
With some restrictions on what a is and y is, of course.
trim fable
they confuse melol
yeah
well i have another question
about domain and range
brb tho
one sec
fluid shore
Try them on your own first
trim fable
i did
i solved for them all
but now just have to do
domain and range and i did it for the first two but
do i use the graph method?
where u draw them out
and set the y and restrict
idk it was confusing i learned it today
fluid shore
Nope, you can do it analytically for all of them. You don't need to draw the graphs, though it will be good practise for you if you do draw them out.
trim fable
oh
so i cant just look at it and
state them?
oh also for example
like one of the examples was
f(x)=x+3 and g(x)=x^2-12
so then what we did was drew g(x) looked at the range and then
fluid shore
Why not just write things down?
fluid shore
??
trim fable
?
fluid shore
Do you know what composition of two functions is?
trim fable
yeah
f(g(x))
but then why are u using g(x) first
when its the inner one?
inner then outer function?
for these?
fluid shore
What do you mean 'why are u using g(x) first'?
fluid shore
Why are you drawing them/
trim fable
and taking its range
fluid shore
?
fluid shore
Why are you drawing them?
trim fable
told us
to draw
them
tho its
confusing
is there a different way to
state domain and range
without having to draw and then
use the range as the domain for f(x)
and then finding it
to see if its valid
fluid shore
Like I said, you can determine it analytically for the given functions. You don't have to draw the individual functions, though it's good practice.
You should actually just google these things. They're not that hard to learn online. Once you have learned them, apply them to your problems and then come back here if you're stuck.
fluid shore
Well, google the composition of functions and how to determine the range/domain of a composite function. They'll give you a whole bunch of results.
fluid shore
Work through the examples you find, then work through your teacher's examples. Then, come back here if you're stuck with an actual step in your working
trim fable
of composites ok thats what i meant
ok
yeah i calculated f(g(x)) and g(f(x))
for all and checked the answrs and got them but
i dd the domain and range for the first 2
tho idk how to do c
coz u get
sqrt(4-x^4)
fluid shore
Okay then?