Did I have it right the first time then?
#help-39
cedar scarab
cedar scarab
Is x bar not 47.4?
cedar scarab
Yee
junior loom
okay, gl, going to lseep
cedar scarab
So $\mu=x\pm \frac{t\sqrt{n}}{s}$?
jolly parrotBOT
MJames
cedar scarab
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pearl pondBOT
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upper comet
pearl pondBOT
tender sleet
mental vine
you use cosine law if:
upper comet
i cant stand wiley
upper comet
like actually gonna crashout
mental vine
- given 2 sides and the angle between them
upper comet
mental vine
alright do them one at a time
the first one:
three sides of a triangle
is that one of the things listed above?
upper comet
HOW TF IS THIS A RIGHT ANGLE
upper comet
schizo homework
upper comet
we have never had a problem not drawn to scale
full agate
Now you have
mental vine
read the question......
omg like 60% of people I help does not read their problem correctly
upper comet
sorry bruh idk whats goin on in this hw
tacit mulch
alternatively, since it seems like you're studying the law of cosines, you can use that to solve for the angle and find that it's 90 degrees
Yeah it can be tempting to trust the picture, but the number one rule of geometry is never assume the image is to scale
upper comet
i dont really understand the law of sines and stuff
we barely glossed over it in class a week ago
.close
pearl pondBOT
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wind lagoon
I'm working an exercise on conic sections. For instance, find new coordinates $x',y'$ so that $x^2+4xy+y^2$ can be written as $\lambda_1(x')^2+\lambda_2(y')^2$. Now, in the text, the authors seem to make a point that this can always be achieved with an orthogonal matrix $P$ with $\det(P)=1$. We can write the equation as $$\begin{pmatrix}x&y\end{pmatrix}\begin{pmatrix}1&2\ 2&1\end{pmatrix}\begin{pmatrix}x\ y\end{pmatrix}=X^tAX.$$Diagonalizing the middleman, we find that $$P^tAP=\begin{pmatrix}-\frac1{\sqrt{2}}&\frac1{\sqrt{2}}\ \frac1{\sqrt{2}}& \frac1{\sqrt{2}}\end{pmatrix}\begin{pmatrix}1&2\ 2&1\end{pmatrix}\begin{pmatrix}-\frac1{\sqrt{2}}&\frac1{\sqrt{2}}\ \frac1{\sqrt{2}}& \frac1{\sqrt{2}}\end{pmatrix}=\begin{pmatrix}-1&0\ 0&3\end{pmatrix}$$ But $P$ is not a rotation, yet it seems that the new coordinates $X'$ are given by $P^tX$ nevertheless. I don't get the point they are making about $P$ being a rotation.
jolly parrotBOT
psie
pearl pondBOT
@wind lagoon Has your question been resolved?
sleek badge
Aight wait
Alright, imagine you're twisting a Rubik's Cube (A) to get all the colors on one side (D). The moves you make are P.
P has to be "orthogonal": This means it's like a solid set of moves – rotations or reflections, no squishing. Its columns are like perfectly perpendicular handles.
X' = P^T X: This is always how you find your new "view" (X') from the old one (X), using this P. P^T is like the "undo" for P's moves when P is orthogonal.
Rotation vs. Reflection:
P can be a pure spin (rotation, det(P)=1).
P can be a spin and a flip (reflection, det(P)=-1). Your P is this type.
The authors' point: They're saying, "Hey, even if your first P is a reflection, you can always tweak it (like flip one of its 'handles') to make it a pure rotation and it'll still solve the cube."
Bottom line: Your P works fine for finding X' because it's orthogonal and does the diagonalization. The authors are just saying you could have chosen a P that was a pure rotation, if you wanted to. The math X' = P^T X doesn't care if P is a spin or a spin-and-flip, as long as it's the right orthogonal P.
wind lagoon
Alright, imagine you're twisting a Rubik's Cube (A) to get all the colors on one...
thanks for this explanation, appreciate it 🙂
swift spindle
Hi how do you solve this
pearl pondBOT
Hi how do you solve this
toxic lichen
swift spindle
toxic lichen
third line should have dy/dx and not y
you shouldn't conflate a function with its derivative
you want to find x such that y=0, and then evaluate dy/dx at that x
swift spindle
The third line I didn’t derive though ?
toxic lichen
or wait
yeah you didn't take any derivatives yet mb
guess im too used to seeing students make this notational mishap lmao
anyway
you overcomped it still
(2x-4)/x^2=0
solve from there
toxic lichen
why wouldn't you be
swift spindle
Ok so x = 2?
toxic lichen
the form y = 2x^-1 - 4x^-2 will be useful for finding dy/dx
toxic lichen
"let f(x) = (2x-4)/x^2
no
toxic lichen
How do you end up subbing 2 in then
x = 2: dy/dx = ...
i mean if you wanna call it f(x) there's nothing wrong w that
pearl pondBOT
@swift spindle Has your question been resolved?
high elbow
how should i go about solving this?
pearl pondBOT
how should i go about solving this?
high elbow
Kernel?
inland ivy
The nullspace
high elbow
Sorry i'm a little lost should I be finding what A-B is as a start?
high elbow
I'm new to SVD, could someone walk me through this problem?
pearl pondBOT
I'm new to SVD, could someone walk me through this problem?
plush bramble
I'm new to SVD, could someone walk me through this problem?
high elbow
Tyty
midnight haven
hey can someone help me learn exponential equations
pearl pondBOT
hey can someone help me learn exponential equations
young oak
try textbooks!!
midnight haven
https://www.khanacademy.org/math/algebra2/x2ec2f6f830c9fb89:exp/x2ec2f6f830c9fb8...
thank you
astral charm
i have a triangle, its point are at 0,0 1,-2 x,y i need to find x and y i know these angles (from right to angle to x,y)
pearl pondBOT
i have a triangle, its point are at 0,0 1,-2 x,y i need to find x and y i know t...
Please show the original problem, exactly as it was stated to you, with the entire original context. A picture or screenshot is best. If the original problem is not in English, then post it anyway! The additional context might still be helpful. Do your best to provide a translation.
astral charm
ok
can i send video
here is the original problem
calculating player's position with 2 eyeballs
eyeballs give angle to player from x axis
steep saddle
uh what
shouldnt it be the other way around where the eyeballs rotation is defined by the player position relative to the eyeballs
why do u got it the other way around
astral charm
shouldnt it be the other way around where the eyeballs rotation is defined by th...
isnt that what i said basically?
steep saddle
nope the variables you said are the player position
steep saddle
is this a cell machine mod
astral charm
remake
steep saddle
ok how are you moving the player then?
dont you have variable(s) to keep track of its position and such
astral charm
wdym
steep saddle
do you store state for the player
astral charm
ok how are you moving the player then?
with arrow keys or wasd
steep saddle
ok so when you press W, what happens in the code?
astral charm
do you store state for the player
eyeballs just find the nearest player cell
and look at them
steep saddle
ok so what i dont understand is why you need to solve for the player position when you already have it
astral charm
i dont
astral charm
i dont want to do it with code i want to do it in game
astral charm
im not coding this im making a single purpose calculator ingame that will use the eyeball's outputs
astral charm
i dont know the math for that
astral charm
i dont know the math for that
i guess i can google the math for that then
astral charm
yeah
steep saddle
and two variables
so that's perfect
just solve the system
use cramers rule if you want
stuck helm
So i am fully confused and stuck with it here it is my syllabus can anyone tell me how i start and how can i learn all this cuzz exam is near nd i am blank or can say at zero . Also i am telling in higher education i studied bilogy but i have to study maths for reason thatss whyy. If anyone can help me in this or guide me i will be highly appreciated.
plush bramble
!ss
pearl pondBOT
Please post images (such as PNGs or JPGs) of the question rather than other filetypes such as PDFs which have to be downloaded. Non-image downloads can potentially contain viruses or other security risks.
desert birch
plush bramble
that's a question for modmail
plush bramble
yea this is not a math problem
you'll have to find resources one unit or even one topic at a time
start with khan academy for unit 1
pearl pondBOT
@stuck helm Has your question been resolved?
pearl pondBOT
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Available help channel!
eager jewel
y=(|x|-1)/(|x|+1)
pearl pondBOT
y=(|x|-1)/(|x|+1)
eager jewel
i need help in finding the range of this function
eager jewel
x > 0, x < 0
x>=0?
autumn fossil
yeah, sure
eager jewel
i did that
autumn fossil
one thing you should notice is that they are practically the same
eager jewel
one thing you should notice is that they are practically the same
yeah
so symmetric yes?
autumn fossil
yeah so for (x-1)/(x+1) i got range as R-{-1}
you need to be careful, your domain is restricted to x >= 0
eager jewel
you need to be careful, your domain is restricted to x >= 0
oh..how do i find the range then with this restricted domain?
autumn fossil
Probably the easiest way would be to just sketch it
eager jewel
Probably the easiest way would be to just sketch it
oh ok so rectangular hyperbola for (x-1)/(x+1) and we take mirror image about y axis?
autumn fossil
oh ok so rectangular hyperbola for (x-1)/(x+1) and we take mirror image about y ...
why mirror image?
oh, yeah
but you only consider the part of rectangular hyperbola with x >= 0
eager jewel
yeah okay
i have a bit of difficulty sketching rectangular hyperbolas could u help
(y-1)(x+1) = -2
is what im getting
eager jewel
this is probably not the most useful form
oh what form do i get it to
autumn fossil
yeah, thats cool
autumn fossil
(y-1)(x+1) = -2
so this is just yx = -2
but translated
do you know how it's translated?
the sketch can be very, very rough btw
eager jewel
1 unit down and 1 unit right?
autumn fossil
not exactly
it has the opposite effect
1 unit up 1 unit left
so that the asymptotes are x = -1, y = 1
eager jewel
😭 how did i mess up so bad lmao
autumn fossil
try making a sketch now and focus on the values for x >= 0
it should look kinda like this, probably better since you are likely a better artist
eager jewel
it should look kinda like this, probably better since you are likely a better ar...
oh ok yeah im getting something like that
and it is that way because of the - sign right?
autumn fossil
and it is that way because of the - sign right?
yep
eager jewel
and cuts the x axis at -1 and y axis as 1
autumn fossil
and cuts the x axis at -1 and y axis as 1
Yes, that's pretty important
but you got it reversed i think
now the range of the values is just this (the y-values of this to be exact)
eager jewel
it should look kinda like this, probably better since you are likely a better ar...
now for the |x| we can do this?
autumn fossil
now for the |x| we can do this?
Yep, not that you need that but the graph would look like that
the blue part basically
eager jewel
oh ok so the range would be
-1 to something
autumn fossil
and the something...?
eager jewel
for that something x---->infinity
autumn fossil
yeah, it's the horizontal asymptote
which can be computed e.g. by taking that limit
autumn fossil
so the range is [-1,1]
check the brackets
eager jewel
[-1,1)?
eager jewel
ohh ok right
oh btw
let me send the actual question
so i didnt actually need to find the range right? i just needed to see that it wasnt R?
autumn fossil
that function is syntactically wrong
autumn fossil
so i didnt actually need to find the range right? i just needed to see that it w...
yeah
eager jewel
that function is syntactically wrong
yeah the function is just what i sent
okay well thanks a lot dude
.close
karmic sonnet
for the difference between 6 and 15, does he not consider the signs and always puts +
karmic sonnet
notice how he put 9f because thats the difference between 15f and 6f
but does he not consider the sign
according to sign it should be + 21f
desert timber
EF = -FE
pearl pondBOT
@karmic sonnet Has your question been resolved?
primal brook
hi
primal brook
I need help with this y=5cos(1/4(x-3))+3
is saying vertical compression of 5 and horizontal stretch of 1/4, right?
<@&286206848099549185>
what about this y=-1/5x-360 +20
cyan oracle
is saying vertical compression of 5 and horizontal stretch of 1/4, right?
other way around its a vertical stretch of 5 and a horizontal compression of 1/4 (which is a horizontal stretch of 4)
-3 inside means 3 units to the right
and + 3 outside means 3 units up
primal brook
ok
cyan oracle
everything that happens to the x is the reverse of what happens to y
primal brook
what about this y=-1/5x-360 +20
?
cyan oracle
take a guess
cyan oracle
everything that happens to the x is the reverse of what happens to y
look at this again
primal brook
yes
cyan oracle
5f(x) would be a vertical stretch of 5
f(5x) would be a horizontal compression of 5
are you saying its 1/(5x) or 1/5 * x
in your expression
cyan oracle
what about this y=-1/5x-360 +20
is this $$y = -\frac{1}{5x -360} + 20$$
jolly parrotBOT
I can't believe you've done this
primal brook
yes
cyan oracle
ok so this is f(5x-360) + 20 where f(x) is 1/x
primal brook
i'm talking about the transformating
cyan oracle
if the 5 was outside itd be a stretch because its inside its a compression of factor 5
primal brook
ok
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Available help channel!
karmic sonnet
why is the sign in front of 9 MINUS instead of PLUS
pearl pondBOT
why is the sign in front of 9 MINUS instead of PLUS
rough forge
why is the sign in front of 9 MINUS instead of PLUS
x²+6x = x²+6x+9-9
just how to complete the square
karmic sonnet
x²+6x = x²+6x+9-9
Yeah but I thought we get it from doing 3x3
Which is 9 I understand
But should be +9
Because x+3
edgy wren
But should be +9
+9-9 so we don't change the quantity
rough forge
what
edgy wren
And write that number in front
yep but you can't jsut add the 9 randomly
you need to keep balance to your quantity that's why we subtract the 9
karmic sonnet
yep but you can't jsut add the 9 randomly
you need to keep balance to your quan...
I didn't add the 9 randomly though, it was x plus THREE squared and so I squared THREE to get 9
edgy wren
I know but still it's random until the quantity is not changed that's why we subtract the 9
rough forge
x²+6x = x²+6x+9-9 = (x²+6x+9)-9 = (x+3)²-9
like
if it was +9 you would screw up the equality
karmic sonnet
I know but still it's random until the quantity is not changed that's why we sub...
Some guy on yt told me this trick as we won't have to do the full (x²+6x) thing to get 9
random ermine
karmic sonnet
x²+6x = x²+6x+9-9 = (x²+6x+9)-9 = (x+3)²-9
Even if I expand that, I'm gonna end up with two nines, one of +9 and one of - 9
How will I know which one to use?
rough forge
always the positive
karmic sonnet
why is the sign in front of 9 MINUS instead of PLUS
It's not even positive there
It's negative
jolly parrotBOT
rough forge
It's not even positive there
but hä thats not where you at
x²+6x = x²+6x+9-9 = (x²+6x+9)-9 = (x+3)²-9
here you use the +9 to complete the square
umbral osprey
because when you complete the square, you use a number that can help with factorizing that part, it becomes -9 because to obtain (x + 3)^2 - 9 from x^2 + 6x, you would need to factor x^2 + 6x and the easy way to do that is to add a "0", so in this case 9 - 9 would be a good "0" to add since you can factorize it and complete the square,
the -9 comes from the fact that they skipped showing (x^2 + 6x + 9 - 9) and went straight into the factored result
karmic sonnet
Ah. So if I'm not expanding the full thing and just doing square of 3, is there any trick to know the sign?
umbral osprey
It's basically just whatever is convenient to you to work with
as long as both numbers add up to 0
karmic sonnet
It's basically just whatever is convenient to you to work with
Wait what, the answer would've been wrong if +9 was used
karmic sonnet
That would take same amount of time
umbral osprey
x^2 + 6x + 9 for me just ends up being more convenient
so since that's (x + 3)^2, we go back to our original question which has x^2 + 6x which isn't exactly the form we want
but because we already know (or can "guess") that (x+3)^2 has something to do with this, we find that the only thing missing is +9
so we just add a "0", which is +9 - 9
if that helps, sorry if it isn't exactly answering your question
but choosing what number is all based off of what you find is convenient to find a square or factored form
karmic sonnet
Ohhh so if I imagine
(x + 8)²
And expand it so (x+8)(x+8)
Getting x²+8x+8x+64
Thus x²+16x+64
I will need to put -64 to get my unhalved stuff for:
(x+8)² -64
Is this right
pearl pondBOT
@karmic sonnet Has your question been resolved?
wanton lagoon
does this count as the proof is done
pearl pondBOT
does this count as the proof is done
@wanton lagoon Has your question been resolved?
wanton lagoon
<@&286206848099549185>
wanton lagoon
.close
pearl pondBOT
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signal comet
Hullo hullo, how can I simplify this more?
pearl pondBOT
Hullo hullo, how can I simplify this more?
eager jewel
Hullo hullo, how can I simplify this more?
huh dont expand it this way
multiply first row by x second row by y and third row by z
and take out xyz from 3rd column
signal comet
multiply first row by x second row by y and third row by z
see, I can't cus I've been specifically told to use the third column, sorry, I should've mentioned that
eager jewel
see, I can't cus I've been specifically told to use the third column, sorry, I s...
ok then open the brackets
and ull need to group some terms
eager jewel
show what u get after opening the brackets
signal comet
this is what I get when I open the brackets
okay I got a riddiculous idea
what if I take this, convert yz, xz and xy into xyz by dividing and multiplying x, y and z and then taking xyz common?
eager jewel
what does that do
eager jewel
doesnt look like u are any closer
signal comet
so...what do I do
eager jewel
there are similar terms in this such as
x^2z and -y^2z
u can take z out
z(x^2-y^2)
z(x+y)(x-y)
try doing this for the other terms
eager jewel
wait one second
this factorization looks a bit tricky
i think u have to use cyclic properties
eager jewel
multiply first row by x second row by y and third row by z
im unable to help u proceed further with this but try doing this to get the fully factorized form and see how it looks like
signal comet
okay ty yoda
okay, I ended up pulling up the solution, wtf happened here
how are we taking y-z common?
hallow ice
okay, I ended up pulling up the solution, wtf happened here
certainly the second line to the last line is not correct
hallow ice
it's not even the jumping ... it's just not correct. (y-z) is not a factor of the second line
It is a factor of the first line though so definitely it can be factored the form (y-z)[something]
signal comet
i see
hallow ice
yz(z-y) let's not bother factorising that because we're planning to take out the (z-y) factor anyway
So let's focus on the last two terms
zx(x-z)+xy(y-x)
can you rearrange that so you write it in the form (y-z)(something)?
signal comet
okay uh let's seee
if I open the brackets, I get zx^2 - z^2x + xy^2 - x^2y
and if I rearrange that
xy^2-z^2x + zx^2 - x^2y
x(y+z)(y-z) - x^2(y-z)
oh hey
nice
hallow ice
hell yeah that looks reasonable
pearl pondBOT
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Available help channel!
desert cliff
could somoene help me out
pearl pondBOT
could somoene help me out
No need to ask “Can I ask…?” or “Does anyone know about…?”—it’s faster for everyone if you just ask your question! See https://dontasktoask.com/
hallow ice
can you solve |x|=0? (this will help you understand the above two lines)
desert cliff
x is equal to 0
hallow ice
yes exactly
so if you see 0=|something| you know that "something" has to be 0
in this case "something" is?
hallow ice
that's it!
pearl pondBOT
pearl pondBOT
✅
desert cliff
which arm do we use
the left or right and why?
im talking about the arm of the absolute value line
hallow ice
it depends what point you care about. you can see the right arm is the relevant one for all points with x>=-2
so if you're doing a computation to do with (0,0) you see the x-coordinate is greater than -2 so the relevant branch is the one with positive slope
but basically, it just comes down to look at the graph: is the point on the arm with postive slope or negative slope?
mighty pond
is that 8th grade math?
pearl pondBOT
@desert cliff Has your question been resolved?
pine steeple
need help
signal patrol
,rotate 180
jolly parrotBOT
sage quartz
You've found g(x) right?
pine steeple
gx is just 6x + 1
sage quartz
That's not quite right
pine steeple
hiow is 6x + 1 even the image'
sage quartz
g(f(x)) maps x to 6x+1
pine steeple
shouldnt it be the relation
sage quartz
Not g(x)
signal patrol
hiow is 6x + 1 even the image'
g maps 3x-2 to 6x+1
sage quartz
hiow is 6x + 1 even the image'
It's the image of the domain element 3x-2 under a certain function g
sage quartz
shouldnt it be the relation
The relation here is g
6x+1 is just one element of the range of g
signal patrol
so ur saying find gfx?
have you found g(x)
pine steeple
yes
signal patrol
find g(x) first
pine steeple
18x -11
pine steeple
i use composite
but this is an exam paper the question thats asking gx
the answer is 6x + 1
its correct
signal patrol
I'm getting confused here
sage quartz
g maps 3x-2 to 6x+1 rught?
pine steeple
yes
pearl pondBOT
@pine steeple Has your question been resolved?
pine steeple
<@&286206848099549185>
pine steeple
here
pine steeple
Whats the general form of a linear function
y = mx + c
i dont understand how u find gx
compact ridge
i dont understand how u find gx
so 3(mx + c) + 2 = 6x + 1
pine steeple
yea
compact ridge
(3mx) + (3c + 2) = (6x) + (1)
pine steeple
(3mx) + (3c + 2) = (6x) + (1)
yea
pine steeple
ig
compact ridge
what this means is that the number in front of the x has to be the same, for both sides to be equal
same for the constant terms on both sides
compact ridge
what are u looking for with this
you want to find m and c right
pine steeple
yea but 3x - 2 is not equal to 6x + 1 tho
compact ridge
yea but 3x - 2 is not equal to 6x + 1 tho
so?
pine steeple
so 3(mx + c) + 2 = 6x + 1
oh i see now
so ur finding the relation
m = 2?
c = -1 /3
compact ridge
c = -1 /3
yeah, well because you know that x will go to mx + c under the function g
so 3(x) will go to 3(mx + c)
3(x) + 2 will go to 3(mx + c) + 2
but we want 3(mx + c) + 2 to equal 6x + 1, cause of the q
compact ridge
nice work
compact ridge
oh well, you know that fg(x) means f(g(x)) right
so that means you do g first not f first
so f(2x - 1/3) = 3(2x - 1/3) + 2 is fg(x)
compact ridge
when its saying fgx
I don't understand
pine steeple
the value of x such that fgx = 4x - 3
compact ridge
or x as in the x inside gx
what do you mean by these two things
pine steeple
ok wait lemme expand firs
tmaybe after i solve i can understand
so its 6x + 1 right
after expansion
compact ridge
yeah
compact ridge
ok ok now fgx is 6x + 2
6x + 1
pine steeple
yea
compact ridge
so what you've done is replace fg(x) by 6x - 1
and the original question was fg(x) = 4x - 3....
pine steeple
6x + 2 = 4x - 3?
compact ridge
6x + 2 = 4x - 3?
6x + 1 = 4x - 3
pine steeple
yes
compact ridge
wait 6x + 1 sorry
pearl pondBOT
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buoyant pasture
buoyant pasture
Need some hints
safe siren
Do you know what gcd is? And knowing that say 1529 can be divided by n and leave a remainder, do you know how to write that as an equation? If you did that for each number you might be able to solve for the variables
safe siren
Okay so what if you did that with all the numbers, you would have 3 equations with a common unknown. do you know what you could maybe do next?
buoyant pasture
No idea what I can do@safe siren
weary ledge
Did you write all the 3 equations for mod?
@buoyant pasture can you subtract any 2 of those equations to see what you get?
safe siren
if you have 1529 = 0 (mod n), what does that actually mean. if you have that for all 3 equations, what does that mean for all the equations? how can you link gcd into this?
buoyant pasture
I can link that all equations will be same 0 remainder@safe siren
safe siren
Want another hint?
pearl pondBOT
@buoyant pasture Has your question been resolved?
sharp smelt
Let $\phi$ be a group homomorphism . I'm trying to prove $\phi (1_G)=1_G$
jolly parrotBOT
What a wonderful world !
sharp smelt
so we have $1_G = 1_G \cdot 1_G$, we then have $\phi(1_G \cdot 1_G) = \phi(1_G) \cdot\phi(1_G) $
jolly parrotBOT
What a wonderful world !
sharp smelt
we then have $\phi(1_G)=0_{G'}$ or $\phi(1_G) = 1_{G'}$ , right? where $\phi: G \to G'$
jolly parrotBOT
What a wonderful world !
tropic saddle
from where
sharp smelt
we multiply both sides by $(\phi(1_G))^{-1}$
jolly parrotBOT
What a wonderful world !
inland ivy
wai what book are you using for group theory?
sharp smelt
Artin's Algebra
sharp smelt
I mean judson is a free alternative
inland ivy
oh I have Artin already
sharp smelt
the text seemed a little dry to me idk
It is a bit dry, yea
I have Judon and aluffi incase I want to switch books
sharp smelt
no problem :D
lunar hare
we then have $\phi(1_G)=0_{G'}$ or $\phi(1_G) = 1_{G'}$ , right? where $\phi: G ...
I think this is a great question as we often assume identity goes to identity without proving it… I think I have a way to see that this is the case, let me know if it is helpful! :
tropic saddle
this is like the first thing proved after defining homomorphisms
sharp smelt
I think this is a great question as we often assume identity goes to identity wi...
thanks
Okay, got it. That's it for now. Thanks a lot everyone!
Can I close this now, I think I got it
.close
flint wyvern
am i right with A here?
pearl pondBOT
am i right with A here?
flint wyvern
it was between that and D
sharp smelt
eh, not really, just checking x=0, we find A evaluvates to -1, when it should evaluvate to 1
flint wyvern
oh shoot okay
sharp smelt
is this a test?
flint wyvern
is this a test?
some practice thing
for a test
still have todo it tho
wait so it could be D then
sharp smelt
,w tan(x-π/4)
sharp smelt
hmm, one muute
jolly parrotBOT
Result:
1.5574077246549
I think you'd want to check other options
versed remnant
I think you'd want to check other options
Can't U rewrite the 1 as tan 45? Works easier
sharp smelt
Can't U rewrite the 1 as tan 45? Works easier
you could yea
versed remnant
Becomes (tan 45- tanx)/(1+tan45(tanx)) which is a known formula
flint wyvern
I think you'd want to check other options
wait would it be C
sharp smelt
why do you think that tho
Well, I just recalled the formula for tan(A-B) and tan(A+B) and tried comparing that
,w tan(x+y)
jolly parrotBOT
sharp smelt
@flint wyvern
flint wyvern
Well, I just recalled the formula for tan(A-B) and tan(A+B) and tried comparing ...
ohh i understand it okay
got it
thank you so mch
.close
pearl pondBOT
pearl pondBOT
Available help channel!
flint wyvern
am i right with D here/
pearl pondBOT
am i right with D here/
jolly parrotBOT
Result:
2
$\sqrt{x^2} = 1 \Ra x = \pm 1$
jolly parrotBOT
riemann
flint wyvern
look at the solutions at the bottom
flint wyvern
oh i thought they just left those two out or something
plush bramble
i just used a calculator for this one and it gave me this tho
this is correct. but your choice doesn't match it
flint wyvern
but now that i look at the question it does say "all"
flint wyvern
this is correct. but your choice doesn't match it
but no other choices have the same
plush bramble
evaluate some of the others for a few n
pearl pondBOT
pearl pondBOT
Available help channel!
cinder bane
Given A, B, C, and D such that "A mod B = C mod D" find the smallest integer X >= 1 such that (A + X) mod B = (C + X) mod D is true
How can I drive a formula or an algorithm to solve that?
cinder bane
I have tried using this method:
$$Y = (A + X) \mod (B) = KB + A + X$$
$$KB + A + X = (C + X) \mod (D)$$
$$KB = (C - A) \mod (D)$$
$$K = (C - A) (B_D^{-1}) \mod D = GD + (C - A)(B_D^{-1})$$
Substituting in the first Y equation:
$$Y = B[ GD + (C - A)(B_D^{-1}) ] + A + X$$
$$Y = GBD + B(C - A)(B_D^{-1}) + A + X$$
$$-Y = G(-BD) - B(C-A)(B_D^{-1}) - A - X$$
$$X = G(-BD) + Y - B(C-A)(B_D^{-1}) - A$$
$$X = Y - B(C-A)(B_D^{-1}) - A \mod (-BD) $$
$$X = Y - B(C-A)(B_D^{-1}) - A \mod (BD) $$
jolly parrotBOT
Sherif Player
cinder bane
Where K and G are some integers
and Y is just the either the remainder of A / B or C / D
The problem with this formula is that it just outputs 0 most of the time, and if did not output zero the result isn't the smallest value possible
So what should I do?
pearl pondBOT
@cinder bane Has your question been resolved?
slate tide
slate tide
.close
pearl pondBOT
pearl pondBOT
Available help channel!
cinder bane
I am reopening this question, hoping that maybe someone would know how to do it:
Given A, B, C, and D such that "A mod B = C mod D" find the smallest integer X >= 1 such that (A + X) mod B = (C + X) mod D is true
How can I drive a formula or an algorithm to solve that?
cinder bane
I have tried this:
$$Y = (A + X) \mod (B) = KB + A + X$$
$$KB + A + X = (C + X) \mod (D)$$
$$KB = (C - A) \mod (D)$$
$$K = (C - A) (B_D^{-1}) \mod D = GD + (C - A)(B_D^{-1})$$
Substituting in the first Y equation:
$$Y = B[ GD + (C - A)(B_D^{-1}) ] + A + X$$
$$Y = GBD + B(C - A)(B_D^{-1}) + A + X$$
$$-Y = G(-BD) - B(C-A)(B_D^{-1}) - A - X$$
$$X = G(-BD) + Y - B(C-A)(B_D^{-1}) - A$$
$$X = Y - B(C-A)(B_D^{-1}) - A \mod (-BD) $$
$$X = Y - B(C-A)(B_D^{-1}) - A \mod (BD) $$
jolly parrotBOT
Sherif Player
cinder bane
Where K and G are some integers
and Y is just the either the remainder of A / B or C / D
The problem with this formula is that it just outputs 0 most of the time, and if did not output zero the result isn't the smallest value possible
So Is there another better way to do that?
feral sedge
I am confused
The answer is usually 1, right
And if it isn't, then it has to be lcm(B,D) - min(A,C)?
Is there a counterexample to this?
cinder bane
hmm
let me check that
A = 3, B = 2, C = 4, D = 3
lcm(B,D) = 2 * 3 = 6
min(A,C) = 3
6 - 3 = 3
5 is the answer
I think you meant A in its simplest form, right?
if so then
A = 1, B = 2, C= 1, D = 3
min(A, C) = 1
6 - 1 = 5
yeah
cinder bane
And if it isn't, then it has to be lcm(B,D) - min(A,C)?
but how does that work
feral sedge
The only pairs of A,C values which solve the equation come all in a row
Starting with A=0,C=0 and continuing up to A=C= min(B,D)-1
And this pattern repeats every lcm(B,D) numbers
So if X=1 fails, we need to go around the loop to the start of the next pattern
Also whenever X=1 fails it's because either B=A+1 or D=C+1 so you can also answer lcm(B,D) - min(B,D) + 1
cinder bane
Starting with A=0,C=0 and continuing up to A=C= min(B,D)-1
So for the equation to be true
A should equal C in their simplest form
The maximum value that either can get to is min(B, D) - 1
Then yeah, the pattern repeats, every lcm(B, D) numbers, I understand that now
cinder bane
Also whenever X=1 fails it's because either B=A+1 or D=C+1 so you can also answe...
So that means that the solution is preaty much independent of A and C
Thanks for you help @feral sedge
.close
pearl pondBOT
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Available help channel!
prisma vortex
pearl pondBOT
sonic hound
,rotate
jolly parrotBOT
sonic hound
What is purpose of the point F?
What is the difference between E and F (points)?
Just use trigonometry.
prisma vortex
Just use trigonometry.
we arent given any lengths
prisma vortex
yeah
prisma vortex
The point of intersection of BF and AD?
how would we start approaching this?
my teacher said those 2 lines can help but im still stuck
sonic hound
Can you prove that BE = EF?
prisma vortex
i think theyre congruent triangles
sonic hound
bye
prisma vortex
?
sonic hound
I need to go
GTG
Someone else should help
blissful salmon
i think theyre congruent triangles
Yep, they are congruent triangles
blissful salmon
Is trigonometry allowed for this?
prisma vortex
Is trigonometry allowed for this?
i think so
blissful salmon
i think so
The reason why I am asking is because the angle is not an integer by my working out. But then you put 15 degrees in the picture you provided, which makes me think that my answer is wrong
prisma vortex
The reason why I am asking is because the angle is not an integer by my working ...
im not sure if it is correct because icl my teacher went over the question because he wanted us to do it by ourself but i saw 15 degrees but im not sure if its wrong did u by chance get 14.04 degres? or am i js dumb
blissful salmon
im not sure if it is correct because icl my teacher went over the question becau...
Yeah I got roughly 14.04 degrees too
prisma vortex
im not sure if we did it wrong tho i dont think it is but idrk as well
blissful salmon
im not sure if we did it wrong tho i dont think it is but idrk as well
I’m pretty sure we did it right assuming that AD and BF intersect at E. Did you also get that the angle was || tan^-1(1/4) degrees || ?
prisma vortex
I’m pretty sure we did it right assuming that AD and BF intersect at E. Did you ...
yeah i got that from (1/2)x/2x
blissful salmon
yeah i got that from (1/2)x/2x
Yes, same
Maybe bring it up with your teacher to see what is happening. But the answer should be valid though
prisma vortex
Maybe bring it up with your teacher to see what is happening. But the answer sho...
yeah alr thx for confirming with me i thought i did smt wrong ill ask him tmr
blissful salmon
yeah alr thx for confirming with me i thought i did smt wrong ill ask him tmr
No problem! All good
prisma vortex
.close
pearl pondBOT
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Available help channel!
kind roost
pearl pondBOT
kind roost
Whadahel how do
bitter herald
Whadahel how do
Hint: [
\6\arg{\4zw} = \6\arg z -\6\arg w
]
jolly parrotBOT
bitter herald
yes but what can you conclude from the fact that arg(z) = arg(w) - pi/2?
kind roost
yes but what can you conclude from the fact that arg(z) = arg(w) - pi/2?
Hmmm what significance does this have
bitter herald
z lags behind w by pi/2 units
kind roost
z lags behind w by pi/2 units
yer yer how does this help me
bitter herald
you can assume, WLOG, that z = -iw in that case
tall bone
compact ridge
you can assume, WLOG, that z = -iw in that case
you can only assume that z = -kiw actually, where k is a real number
z and w do not necessarily have the same magnitude
kind roost
Help me south 🙌
bitter herald
you can only assume that z = -kiw actually, where k is a real number
you are right, but luckily you can still do it with that in mind
bitter herald
yer yer, i sub that in?
sub in z = -kiw like south rightfully corrected and simplify your fractionn
compact ridge
you are right, but luckily you can still do it with that in mind
magnitude of (-ki - 1)/(-ki + 1)?
bitter herald
yeah
verbal whale
magnitude of (-ki - 1)/(-ki + 1)?
But don't the two numbers have the same modulus?
compact ridge
But don't the two numbers have the same modulus?
oh lmao yeah that's so true
for some reason I saw they weren't complex conjugates and went on a wild goose chase
but they are reflections of each other across the y-axis
verbal whale
for some reason I saw they weren't complex conjugates and went on a wild goose c...
Me too at the beginning, I really hoped they were conjugates but luckily they still have the same mod
kind roost
I swear this is so ceebs
verbal whale
but they are reflections of each other across the y-axis
Yeah exactly, thankfully the author made it solvable lol
compact ridge
Help me south 🙌
let the apex of this triangle be the origin
that's what the two complex numbers look like graphically
compact ridge
from the origin:
k units down then 1 unit left
k units down then 1 unit right
pearl pondBOT
compact ridge
ah this is very subtly wrong, should be -k^2 instead
jolly parrotBOT
south
verbal whale
Here, and in general, it might help use this property:
$$\abs{\frac{A}{B}} = \frac{\abs{A}}{\abs{B}}$$
jolly parrotBOT
Alberto Z.
verbal whale
Because it simplifies calculations a lot, for instance you don't need to multiply by the conjugate of the denominator as you did
pearl pondBOT
Available help channel!
lucid hazel
pearl pondBOT
lucid hazel
i want to find the rational equation from graph, i just have a question too
is the numerator (x-2)^3?
autumn fossil
i dont think so
have you found the denominator already?
the numerator must have smaller degree, because it approaches 0 at +-infinity
lucid hazel
i dont think so
ill tell you what influenced my answer
this
shows multiplicity of 3?
right?
i remember taking this in class
autumn fossil
shows multiplicity of 3?
it looks almost like it
but not exactly
multiplicity 3 is perfectly flat at that point
this is just nearly flat
lucid hazel
my bad i guess lol
another question
when i look at this graph , i immediately think of 1/x^2 form
and it should be positive , right?
because the -1/x^2 looks down
ok then what is this
😭
i am rlly confused
i mean the HA does say something
but it kinda contradicts
what i know
autumn fossil
because the -1/x^2 looks down
lets go with this idea
you'll eventually end up with what they have
translate it such that the asymptotes are right first
-1/x^2 has asymptotes at y = 0 and x = 0
autumn fossil
another question
this has them at y = -1 and x = 1
autumn fossil
translate it such that the asymptotes are right first
do you know how to do it?
lucid hazel
do you know how to do it?
i dont understand u to be honest
frank goblet
If you apply these transforms to get your f(x), you can simplify into one single fraction, then it will look like the answer
Your 1/x^2 is correct, now shift it to the right and down, then simplify the result into a fraction
lucid hazel
If you apply these transforms to get your f(x), you can simplify into one single...
huh? it will be -x^2 + 2 / (x-1)^2
frank goblet
Almost
When we shift 1 down, we get a -1 at the end of the function. Achieving common denominators, we get -1=-(x-1)^2/(x-1)^2=-(x2-2x+1)/(x-1)^2
So the 1s cancel
lucid hazel
are you able to write that using the bot @jolly parrot
so i can see it
please
cuz that confuses me
frank goblet
$f(x)=\frac{1}{(x-1)^2}-1=\frac{1}{(x-1)^2}-\frac{(x-1)^2}{(x-1)^2}=\frac{1}{(x-1)^2}-\frac{x^2-2x+1}{(x-1)^2}$
jolly parrotBOT
Martin
lucid hazel
thanks
pearl pondBOT
@lucid hazel Has your question been resolved?
eager jewel
pearl pondBOT
eager jewel
i assumed it as a 2 degree polynomial
f(x)=ax^2+bx+c
but im getting everything = 0
pearl pondBOT
@eager jewel Has your question been resolved?
heavy hill
pearl pondBOT
heavy hill
i know it says "In the following questions,
you will need to consider the amount of UV light let through."
but in similar question i have not seen this statement and you can only solve the questions if find equations using light let through
why does it not work when you use light blocked?
earnest finch
but in similar question i have not seen this statement and you can only solve th...
Can you show the similar question?
earnest finch
why does it not work when you use light blocked?
If you stack two cloths on each other, the amount of light let through will reduce.
earnest finch
If you stack two cloths on each other, the amount of light let through will redu...
For example, let's assume we have 2 cloths that each let in 50% of UV rays.
heavy hill
If you stack two cloths on each other, the amount of light let through will redu...
yeah and in the same way if you stack to cloths the amount of light blocked will increase
earnest finch
For example, let's assume we have 2 cloths that each let in 50% of UV rays.
If we stack both of the cloths, do you expect the total light let in to be 0.5 * 0.5 = 0.25? Or do you expect the total light blocked to be (1 - 0.5) * (1 - 0.25) = 0.25?
heavy hill
why do these questions only work when we use light let through and not blocked
earnest finch
why do these questions only work when we use light let through and not blocked
Because you must treat the amount of light let through as a decimal, and multiply those quantities together. Multiplying the light blocked out by each cloth will not help.
heavy hill
because light let through:
50, 25, 12.5 (forms a gp)
light blocked:
50, 75, 87.5 (doesn't form a pattern)
earnest finch
because light let through:
50, 25, 12.5 (forms a gp)
light blocked:
50, 75, 87....
Sorry, what is a gp?
heavy hill
geometric progression
earnest finch
because light let through:
50, 25, 12.5 (forms a gp)
light blocked:
50, 75, 87....
Then it seems like you yourself agree that we must consider the amount of light let through rather than the amount of light blocked.
heavy hill
Then it seems like you yourself agree that we must consider the amount of light ...
yeah but i am confused as to how someone would know to use it and why light blocked won't form a pattern
earnest finch
yeah but i am confused as to how someone would know to use it and why light bloc...
Light Source
----------------- lets in 33% of light
67% of the light```Try to imagine this, with a light source pointing down.If we put a second cloth below the first one, will the second cloth affect any light that doesn't reach it?
earnest finch
no
Exactly, so the second cloth will not reduce (or change at all) any already blocked light.
heavy hill
oh so your saying how the numbers in the amount of light blocked doesn't affect each number, so no series is formed
ohh ok yeah that makes sense
thank you so much!
.close
pearl pondBOT
earnest finch
oh so your saying how the numbers in the amount of light blocked doesn't affect ...
You can technically form a series with something like 1 - r^n.
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Available help channel!
flint wyvern
am i right with D here?
pearl pondBOT
am i right with D here?
earnest finch
am i right with D here?
✅
pearl pondBOT
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Available help channel!
heavy hill
I dont get how its to the power of 15 for the first question, i though it would be to the power of 14
pearl pondBOT
I dont get how its to the power of 15 for the first question, i though it would ...
@heavy hill Has your question been resolved?
lucid wharf
I dont know how to shade this in🧍♀️
pearl pondBOT
I dont know how to shade this in🧍♀️
edgy wren
I dont know how to shade this in🧍♀️
third root of x is not x^(3/2)
lucid wharf
I dont🧍♀️
edgy wren
I dont🧍♀️
ever took inverse functions ? (the easier method)
lucid wharf
what do u mean
edgy wren
like inverse functions like
x^2 is +-sqrt(x) for example
and the inverse function is a function mirrored along the line y=x
lucid wharf
You want me to find the inverse function of x^1/3?
edgy wren
You want me to find the inverse function of x^1/3?
knowing it well help on graphing
simple answer is x^3
edgy wren
Like this
The (x,y) for x^3 becomes
(y,x) in x^1/3
edgy wren
Like this
The (x,y) for x^3 becomes
(y,x) in x^1/3
So in f(x) here (2,8)
In g(x) its (8,2)
That’s how inverse function (mostly) work
edgy wren
then just plot the inputs using a calc or sum then
pearl pondBOT
@lucid wharf Has your question been resolved?
sour hull
edgy wren
another help room
sour hull
Ok
lucid wharf
So in f(x) here (2,8)
In g(x) its (8,2)
im watching a video and what im getting is since y = x^1/3 and were in the range 0 to 8 do i do y=(0,1,2,3,4,5,6,7,8)^1/3 points = (0,0) ,(1,1),(2,1.2),(3,1.4),(4,1.5) and so on? im just plugging in to find where to shade under?
edgy wren
im watching a video and what im getting is since y = x^1/3 and were in the range...
idea is correct values are wrong
(0,0)
(1,1)
(2,2^1/3)
.
.
.
(8,2)
and do it for the negative part too
pearl pondBOT
@lucid wharf Has your question been resolved?
stoic imp
pearl pondBOT
stoic imp
need help c) and d) mates