#Can someone help me graph this one?
shell wraithBOT
warped kestrel
Ohh sorry, I'm just trying to edit my text but can't do it, so i thought of just deleting it and making a new post
clever sinew
already it looks like theres a typo, this should be forall a, not forall x
warped kestrel
Ohh yeah
clever sinew
needless to say theres a lot of requirements scattered around here
now youre allowed to use a piecewise function, right
warped kestrel
Wdym?
clever sinew
have you learned what piecewise functions are
,,\begin{cases}-1&\text{if }x<0\1&\text{if }x\ge0\end{cases}
timber mauveBOT
mtt
clever sinew
they look like this
warped kestrel
Yeah, i know what piecewise are
clever sinew
alr, so we'll have to use some piecewises for each particular section
what we're looking for is just any easy function that can do whats required for each one
then the piecewise format can put them all together
first, we can consider (-inf, -1)
try coming up with a function where
- as x -> -inf, the function -> inf
- the function is differentiable
- the function is decreasing
the function needs to at least do that in (-inf, -1), the piecewise format will only look there
ignore the problem for now
clever sinew
try coming up with a function where
* as x -> -inf, the function -> inf
* the fu...
just tell me a function that does this
warped kestrel
Function? U mean make one to satisfy that? Is that necessary? I essentially tasked to just make a one graph given those conditions, I'm just struggling where to start.
clever sinew
cmon you gotta place some trust in me
and that begins by telling me an example of a function that does this, I can promise you it is useful for later on
you gotta communicate if you dont want to do it this way
do you want to do something else?
warped kestrel
Sorry sorry, im just not quite sure how to do it
clever sinew
here, lets try something
lets say I give you three requirements, and you tell me an example of a function that fits those three requirements
warped kestrel
I'll try
clever sinew
its separate from the problem for now, alright?
here are the three requirements
- as x -> -inf, the function -> inf
- the function is differentiable
- the function is decreasing
do you need help with any of these requirements?
warped kestrel
I think a negative cubic
warped kestrel
Ohh yeah
clever sinew
yea
thats all we need out of a function like this, straight lines will work for most of these sections
now heres the next set of requirements
- differentiable
- as x -> -1, f(x) -> 1
- as x -> -inf, f(x) -> inf
- f(0) = 0
- as x -> 0^-, f(x) -> 0
as you can see, theres more requirements
what function can do all of these requirements?
and again, dont overthink this
warped kestrel
Is that a piecewise? I can't really think of any
clever sinew
have you tried y = -x a second time?
warped kestrel
Oh crap
clever sinew
cmon man
warped kestrel
Yeah mb
clever sinew
so already -x can do all of that
ok this time these are different requirements:
- continuous
- as x -> 0^+, f(x) -> 1
- f(1) = -1
- as x -> 1^-, f(x) -> -1
as before, try doing something simple like y = mx + b
warped kestrel
-2x+1?
clever sinew
yep, thats what I got
heres another:
- continuous
- f(1) = -1
- as x -> 1^+, f(x) -> -1
- as x -> 2^-, f(x) -> 0
(this is not -x and not -2x + 1 btw)
warped kestrel
Is this still simple function? What i can only think of is just a piecewise of x - 2 between 1 and 2, and 0 for >=2
clever sinew
heres another:
* continuous
* f(1) = -1
* as x -> 1^+, f(x) -> -1
* as x -> 2^-,...
the function y = x - 2 satisfies these requirements
do you understand that
warped kestrel
I thought im supposed to find another function, im fogging my brain up, sorry
clever sinew
well you are finding another function
that function for example is y = x - 2
instead of having to be "y = x - 2 for 1 <= x < 2 and y = 0 for x >= 2"
again you can put your trust in me
you dont have to do the entire piecewise part by yourself, otherwise you wouldve done that by now
one step by a time alright?
warped kestrel
Yeah sure
clever sinew
now this is a bit harder to pull off
try these requirements:
- continuous for x not equal to 3 (to make things easier to consider)
- as x -> 2^+, f(x) -> 0
- as x -> 3^-, f(x) -> inf
a line wont work for this one, but theres still many things that can fit these requirements
warped kestrel
Idk, probably 1/3-x?
clever sinew
at x = 2, 1/(3-2) would be 1 instead of 0
theres a simple change you can do to 1/(3-x) to have it work out
warped kestrel
Have numerator to be x-2? Or I'm overthinking this
clever sinew
thats interesting, I hadnt thought of that
that also works, its just as simple
I was thinking to do 1/(3 - x) - 1
that and this should be the same function
so (x-2)/(3-x) works for these requirements
alr last set of requirements
- continuous for x >= 3 (this can still mean its continuous for x >= 2, x >= 1, or whatever, it just at least has to be continuous for x >= 3)
- f(3) = 0
- as x -> inf, f(x) -> 2
warped kestrel
Oh, 2(x-3)/x-2?
clever sinew
thats interesting, but you dont need the (x - 2) in the bottom, just x works
remember to put in parentheses
warped kestrel
Ohh yeah
clever sinew
alr, so we now have a big pile of functions that satisfy some but not all of the requirements
warped kestrel
Sure
clever sinew
of all the requirements that we have,
these are the functions that we ended up placing
all thats left is to figure out a piecewise that can be the correct function for the correct values of x
now heres the piecewise that we're working with:
[f(x)=\begin{cases}&\text{if }x<-1\&\text{if }x=-1\&\text{if }-1<x\le0\&\text{if }0<x\le1\&\text{if }1<x<2\&\text{if }2<x<3\&\text{if }x\ge3\end{cases}]
timber mauveBOT
mtt
clever sinew
the idea here is that, in each case of the piecewise (each row), we write down a function, so that f(x) should fit every requirement
now conveniently Ive already sorted the requirements by interval then asked you about how to fill each one of them in
so for example, what function would go up here
warped kestrel
Isn't that just the -x
clever sinew
yep
[f(x)=\begin{cases}-x&\text{if }x<-1\&\text{if }x=-1\&\text{if }{-1}<x\le0\&\text{if }0<x\le1\&\text{if }1<x<2\&\text{if }2<x<3\&\text{if }x\ge3\end{cases}]
timber mauveBOT
mtt
clever sinew
now knowing that, try filling in the rest of the piecewise
if youre ever unsure, you can look back to the requirements to check that the function works/doesn't work
warped kestrel
I'll try, brb
clever sinew
alr, gl
warped kestrel
these are the functions that we ended up placing
I just realized it's already in here
clever sinew
thats correct
technically, thats what youve already done so far, except for the dot colors and the 2 in the upper-left corner
warped kestrel
So with the top conditions, some are just some removable discons?
I was also tasked to list the x's discons, where it's not diff., interval on where they are diff., and domains
I'll try to answer it and I'll let you check it
clever sinew
So with the top conditions, some are just some removable discons?
back
if you go for piecewise, you dont have to worry about removable discontinuities, since youre just manually writing down what each portion should/shouldn't do
removable discontinuities in this context would mean doing a (x-2)/(x-2) or something to prevent a function from being valid at 2 but changing nothing else
since we're using piecewises, we dont have to resort to that
we could just redefine the function at x=2 to for example not exist
you can see we have something like that for x = -1
warped kestrel
Ohh i see, i just wanna ask if the domain would be from -inf, inf? I cant see any restrictions
Domain of the whole function
clever sinew
Ohh i see, i just wanna ask if the domain would be from -inf, inf? I cant see an...
the domain is every number except 2, since the function needs f(2) to be DNE
warped kestrel
Ohh yeah the asymptote. Btw thanks man