anonymous
  • anonymous
TuringTest, functions?
Mathematics
  • Stacey Warren - Expert brainly.com
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chestercat
  • chestercat
I got my questions answered at brainly.com in under 10 minutes. Go to brainly.com now for free help!
TuringTest
  • TuringTest
Right, so now that we've covered a lot of that let's look for something tricky: starting simple: y=x+7 is y a function of x?
anonymous
  • anonymous
Yes
TuringTest
  • TuringTest
good now, is x a function of y ?

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anonymous
  • anonymous
No
anonymous
  • anonymous
Cause x is the dependent variable
TuringTest
  • TuringTest
x is the independent variable, but the independent variable can be a function of the dependent variable sometimes. It can work both ways. Not always, though. You must check. to find out try solving for x above
anonymous
  • anonymous
y=x+7 y-7=x?
TuringTest
  • TuringTest
so is x a function of y?
TuringTest
  • TuringTest
can we switch their places?
anonymous
  • anonymous
They should work both ways.
TuringTest
  • TuringTest
right, so in y=x+7 we can write one variable as a function of the other at will x=y-7 so we can change the way we look at this and call it f(x)=x+7=y or g(y)=y-7=x they are both functions and both equivalent. now trickier: what about y=x^2 is y a function of x ?
anonymous
  • anonymous
Yes?
TuringTest
  • TuringTest
yes, each x gives exactly one y now the hard part :P is x a function of y?
anonymous
  • anonymous
No, cause each y gives us 2 x
anonymous
  • anonymous
It could be a function of yx
TuringTest
  • TuringTest
how so? elaborate ;)
anonymous
  • anonymous
|dw:1326051567892:dw| every element of a is maped to an element of b
anonymous
  • anonymous
y=x^2 yx=x Therefore each y gives us 2 x It is a function of yx
TuringTest
  • TuringTest
nice try but not, the algebra is wrong btw zbay's drawing show essentially the same concept as the tables I made y=x^2 solve for x do you know how?
anonymous
  • anonymous
y/x=x actually
anonymous
  • anonymous
Lol i rushed it
TuringTest
  • TuringTest
that's not solved for x, you have x on both sides :/
TuringTest
  • TuringTest
take the square root of both sides
anonymous
  • anonymous
y1/2=x?
TuringTest
  • TuringTest
is the 1/2 and exponent?
TuringTest
  • TuringTest
an exponent*
anonymous
  • anonymous
Yes, didnt you say that yesterday?
TuringTest
  • TuringTest
yes, just making sure, you should write it as y^(1/2)=x to avoid confusion that is right but you forgot one little detail about taking a square root of both sides of an equation...
anonymous
  • anonymous
ALright
TuringTest
  • TuringTest
plus or minus...
anonymous
  • anonymous
What?
TuringTest
  • TuringTest
when you solve\[x^2=4\]there are two answers, remember? one positive and one negative. or did you not know that?
anonymous
  • anonymous
No, why?
pokemon23
  • pokemon23
TURNING TEST BUDDY :D
TuringTest
  • TuringTest
look at two different ways of getting the answer:\[x^2=y\]look at y=4:\[2^2=4\]\[(-2)^2=4\]so x=2 and x=-2 BOTH correspond to y=4 This is fine for y as a function of x, but means that x cannot be a function of y, because one y corresponds to multiple x.
TuringTest
  • TuringTest
hi pokemon :)
anonymous
  • anonymous
ooooh
anonymous
  • anonymous
since negative multiplied by negative is positive
TuringTest
  • TuringTest
exactly :D
TuringTest
  • TuringTest
so whenever you have to take the square root like that we must write\[x^2=y\to x=\pm\sqrt y\]which shows that x is not a function of y sometimes we can ignore this and only look at the positive root to make it a function, but we may have to adjust things accordingly to do that. Here is something many tutors on Open Study still don't know so listen up...
TuringTest
  • TuringTest
If you are asked What is \[\sqrt4\]there is only one answer, 2 but if you are are asked \[x^2=4\]what is x? you must look at both plus and minus:\[x^2=4\to x=\pm2\] Many tutors here think that you always need +/- when you look at a square root. The truth is you only do that when you have to TAKE the square root of both sides of an equation. There, be a step ahead ;)
TuringTest
  • TuringTest
so that said, back on topic\[r^2=t\]is t a function of r? is r a function of t?
anonymous
  • anonymous
r is a function of t but not vice versa
TuringTest
  • TuringTest
actually vice versa, but not vice versa lol
TuringTest
  • TuringTest
which is a function of what? careful.
anonymous
  • anonymous
So t is a function of r?
TuringTest
  • TuringTest
yes, because it's okay for t to be the dependent variable. one r gives one t If we turn it around and solve for r however, we have to use +/-sqrt so that means r won't be a function of t.\[r^2=t\to r=\pm\sqrt t\]so each value of t except zero will give us two real answers. not a function
anonymous
  • anonymous
Ohh
TuringTest
  • TuringTest
each positive value of t will give two real answers*
anonymous
  • anonymous
Yeah, throw me another one
TuringTest
  • TuringTest
what about\[r=\sqrt t\]???
anonymous
  • anonymous
sqrt(t) cant be the dependent variable
TuringTest
  • TuringTest
actually it can because I took away the +/- sign remember what I said about how many tutors are wrong about square roots? in the case of\[\sqrt4=2\]we have one answer, but for\[x^2=4\to x=\pm2\]we have two answers. so this is equivalent to the first case, we are only looking at the /positive/ square root.
anonymous
  • anonymous
Oh, so if the +/- sign is gone then it can me the other way around?
TuringTest
  • TuringTest
it is the act of TAKING the square root that introduces the +/-, which is what makes it not a function. If we don't have to actually take the square root, then it is just positive. so\[r=\sqrt t\]is a function
TuringTest
  • TuringTest
yes because we are only looking at the positive root I'll draw...
TuringTest
  • TuringTest
I made our graph with r as the vertical direction so we could use the vertical line test. Think of how that works.|dw:1326053937990:dw|I'll draw in r=sqrt(t)...
TuringTest
  • TuringTest
Actually first the graph of r^2=t:|dw:1326054064202:dw|
TuringTest
  • TuringTest
Now the graph of just\[r=\sqrt t\]|dw:1326054150478:dw|see how we just use the positive part? notice how r^2=t does not pass the vertical line test with t as the independent variable, so r is not a function of t however in our last graph we clearly do have r=sqrt(t) with r as a function ot t.
TuringTest
  • TuringTest
|dw:1326054308550:dw|^not a function|dw:1326054323340:dw|^is a function
anonymous
  • anonymous
because of the plus/minus it can be |dw:1326054372813:dw| or |dw:1326054351716:dw| Depending on if its negative or positive
TuringTest
  • TuringTest
but that would be splitting the graph up from it's original form if we start Let me try to convey the idea graphically and with the table...
anonymous
  • anonymous
So that was wrong?
TuringTest
  • TuringTest
Nothong really I just want to clear up the idea of what it means to break up the function like that. say we have\[y=x^2\]|dw:1326054620838:dw|you would agree that y is a function of x, yes?
anonymous
  • anonymous
No its not
TuringTest
  • TuringTest
because it passes the vertical line test|dw:1326054781944:dw|but what if we changed the coordinates on the graph put x vertical and y horizontal|dw:1326054851735:dw|even though the function hasn't changed, by changing the axis and using the vertical line test we show that y is a function of x, but not vice versa
anonymous
  • anonymous
ut its x, not y. Shouldnt it be horizontal line test?
TuringTest
  • TuringTest
no, vertical line test for a graph of independent variable on the horizontal dependent on the vertical however doing a horizontal line test would amount to the same thing as changing the axes if you noticed ;)|dw:1326055073462:dw|same result, we see that y is a function of x, but not the other way around.
TuringTest
  • TuringTest
so another way to see it, is|dw:1326055195330:dw|is y a function of x?
anonymous
  • anonymous
Yes
TuringTest
  • TuringTest
good is x a function of y?
anonymous
  • anonymous
no
TuringTest
  • TuringTest
show please
anonymous
  • anonymous
|dw:1326055271855:dw|
TuringTest
  • TuringTest
perfect :)
anonymous
  • anonymous
:D
TuringTest
  • TuringTest
last thing is easier, tables... x | y ---- 2 | 2 5 | 1 0 | 3 4 | 6 is y a function of x ? is x a function of y ?
anonymous
  • anonymous
y is a function of x and x is a function of y
TuringTest
  • TuringTest
excellent actually I should be more careful with my language. With the table we can only say that this is potentially a function because we don't necessarily have all the values. That's a technicality though. ok last one...
TuringTest
  • TuringTest
r | t --- 0 | 1 2 | 3 5 | 9 7 | 3 4 | -14 is r(t) a function? is t(r) a function?
anonymous
  • anonymous
No Yes
TuringTest
  • TuringTest
correct! very nice, I think you understand the concept of functions :D
anonymous
  • anonymous
Seriously? :D
TuringTest
  • TuringTest
yes, you are right :D care to state your rationale?
anonymous
  • anonymous
Rationale?
TuringTest
  • TuringTest
why? your reason.
TuringTest
  • TuringTest
how did you know?
anonymous
  • anonymous
Oh, for the answer?
TuringTest
  • TuringTest
yeah r | t --- 0 | 1 2 | 3 5 | 9 7 | 3 4 | -14 is r(t) a function? is t(r) a function WHY?
anonymous
  • anonymous
r | t --- 0 | 1 2 | 3<-\ 5 | 9 These make t have several values of the same variable (2,7) I dont see 7 | 3<-/ duplicates on the other side. 4 | -14
TuringTest
  • TuringTest
right, so r(t) does not represent a function, where t(r) does perfect!!!!
anonymous
  • anonymous
Yeah!!! :DDD
TuringTest
  • TuringTest
so now you have three interpretations of a function: graphical, numerical, and mathematical. try to harmonize them in your mind, they will be your best friends :)
anonymous
  • anonymous
Ok :D Im going to take a 30min break now :)
TuringTest
  • TuringTest
well deserved, take your time :D

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