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TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.3Right, 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?

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.3good now, is x a function of y ?

Inopeki
 2 years ago
Best ResponseYou've already chosen the best response.0Cause x is the dependent variable

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.3x 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

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.3so is x a function of y?

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.3can we switch their places?

Inopeki
 2 years ago
Best ResponseYou've already chosen the best response.0They should work both ways.

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.3right, so in y=x+7 we can write one variable as a function of the other at will x=y7 so we can change the way we look at this and call it f(x)=x+7=y or g(y)=y7=x they are both functions and both equivalent. now trickier: what about y=x^2 is y a function of x ?

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.3yes, each x gives exactly one y now the hard part :P is x a function of y?

Inopeki
 2 years ago
Best ResponseYou've already chosen the best response.0No, cause each y gives us 2 x

Inopeki
 2 years ago
Best ResponseYou've already chosen the best response.0It could be a function of yx

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.3how so? elaborate ;)

zbay
 2 years ago
Best ResponseYou've already chosen the best response.0dw:1326051567892:dw every element of a is maped to an element of b

Inopeki
 2 years ago
Best ResponseYou've already chosen the best response.0y=x^2 yx=x Therefore each y gives us 2 x It is a function of yx

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.3nice 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?

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.3that's not solved for x, you have x on both sides :/

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.3take the square root of both sides

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.3is the 1/2 and exponent?

Inopeki
 2 years ago
Best ResponseYou've already chosen the best response.0Yes, didnt you say that yesterday?

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.3yes, 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...

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.3when you solve\[x^2=4\]there are two answers, remember? one positive and one negative. or did you not know that?

pokemon23
 2 years ago
Best ResponseYou've already chosen the best response.1TURNING TEST BUDDY :D

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.3look 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.

Inopeki
 2 years ago
Best ResponseYou've already chosen the best response.0since negative multiplied by negative is positive

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.3so 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
 2 years ago
Best ResponseYou've already chosen the best response.3If 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
 2 years ago
Best ResponseYou've already chosen the best response.3so that said, back on topic\[r^2=t\]is t a function of r? is r a function of t?

Inopeki
 2 years ago
Best ResponseYou've already chosen the best response.0r is a function of t but not vice versa

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.3actually vice versa, but not vice versa lol

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.3which is a function of what? careful.

Inopeki
 2 years ago
Best ResponseYou've already chosen the best response.0So t is a function of r?

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.3yes, 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

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.3each positive value of t will give two real answers*

Inopeki
 2 years ago
Best ResponseYou've already chosen the best response.0Yeah, throw me another one

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.3what about\[r=\sqrt t\]???

Inopeki
 2 years ago
Best ResponseYou've already chosen the best response.0sqrt(t) cant be the dependent variable

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.3actually 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.

Inopeki
 2 years ago
Best ResponseYou've already chosen the best response.0Oh, so if the +/ sign is gone then it can me the other way around?

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.3it 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
 2 years ago
Best ResponseYou've already chosen the best response.3yes because we are only looking at the positive root I'll draw...

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.3I made our graph with r as the vertical direction so we could use the vertical line test. Think of how that works.dw:1326053937990:dwI'll draw in r=sqrt(t)...

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.3Actually first the graph of r^2=t:dw:1326054064202:dw

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.3Now the graph of just\[r=\sqrt t\]dw:1326054150478:dwsee 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
 2 years ago
Best ResponseYou've already chosen the best response.3dw:1326054308550:dw^not a functiondw:1326054323340:dw^is a function

Inopeki
 2 years ago
Best ResponseYou've already chosen the best response.0because of the plus/minus it can be dw:1326054372813:dw or dw:1326054351716:dw Depending on if its negative or positive

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.3but 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...

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.3Nothong 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:dwyou would agree that y is a function of x, yes?

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.3because it passes the vertical line testdw:1326054781944:dwbut what if we changed the coordinates on the graph put x vertical and y horizontaldw:1326054851735:dweven 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

Inopeki
 2 years ago
Best ResponseYou've already chosen the best response.0ut its x, not y. Shouldnt it be horizontal line test?

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.3no, 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:dwsame result, we see that y is a function of x, but not the other way around.

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.3so another way to see it, isdw:1326055195330:dwis y a function of x?

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.3good is x a function of y?

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.3last 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 ?

Inopeki
 2 years ago
Best ResponseYou've already chosen the best response.0y is a function of x and x is a function of y

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.3excellent 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
 2 years ago
Best ResponseYou've already chosen the best response.3r  t  0  1 2  3 5  9 7  3 4  14 is r(t) a function? is t(r) a function?

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.3correct! very nice, I think you understand the concept of functions :D

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.3yes, you are right :D care to state your rationale?

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.3yeah r  t  0  1 2  3 5  9 7  3 4  14 is r(t) a function? is t(r) a function WHY?

Inopeki
 2 years ago
Best ResponseYou've already chosen the best response.0r  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
 2 years ago
Best ResponseYou've already chosen the best response.3right, so r(t) does not represent a function, where t(r) does perfect!!!!

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.3so 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 :)

Inopeki
 2 years ago
Best ResponseYou've already chosen the best response.0Ok :D Im going to take a 30min break now :)

TuringTest
 2 years ago
Best ResponseYou've already chosen the best response.3well deserved, take your time :D
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