can anyone cross-check this for me and help me explain the last one | Mathematics | Page 1

smoky plover Sep 11, 2023, 2:09 PM

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Consider the equation and the relation “(x, y) R (0, 2)”, where R is read as “has distance 1 of”.
For example, “(0, 3) R (0, 2)”, that is, “(0, 3) has distance 1 of (0, 2)”. This relation can also be read as “the point (x, y) is on the circle of radius 1 with center (0, 2)”. In other words: “(x, y) satisfies this equation , if and only if, (x, y) R (0, 2)”.
Does this equation determine a relation between x and y? Can the variable x can be seen as a function of y, like x=g(y)? Can the variable y be expressed as a function of x, like y= h(x)? If these are possible, then what will be the domains for these two functions? What are the graphs of these two functions?
Are there points of the coordinate axes that relate to (0, 2) by means of R?
Your Discussion should be a minimum of 250 words in length and not more than 750 words.

According to the question above, it says, does the equation above determine a relation between x and y?
Answer:
Yes, it does. The relation between x and y is y=3-x
Prove:
x2 +(y-2)2 =1
(y-2)2 = 1- x2
(y-2)2 = 12- x2 i.e. (1- x)2
(y-2)2 = (1- x)2
y-2 = 1-x
y=1-x+2
y=3 - x

Can the variable x can be seen as a function of y, like x=g(y)?
Yes
Prove:
y=3 - x
x= 3-y

Can the variable y be expressed as a function of x, like y= h(x)?
Yes

Prove:
y=3 - x

If these are possible, then what will be the domains for these two functions? What are the graphs of these two functions?
can anyone cross-check this for me and help me explain the last one

somber sedgeBOT Sep 11, 2023, 2:09 PM

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smoky plover Consider the equation and the relation “(x, y) R (0, 2)”, where R is read as “...

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amber pike Sep 11, 2023, 3:30 PM

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Wait I'm really confused

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So are you asking if the circle of radius 1 with origin at (0,2) can be expressed as a relation between x and y?

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If yes then it's x^2 (y-2)^2 = 1

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Also did you mean to write x^2 and (y-2)^2 instead of x2 and (y-2)2?

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Ahh ok I think I see that the problem is

amber pike Sep 11, 2023, 3:32 PM

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amber pike If yes then it's x^2 (y-2)^2 = 1

I see you did that right but you just used other notations

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The problem is when you wrote (y-2)^2 = 1 - x^2

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and it is true that 1 = 1^2

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But 1^2 - x^2 =/= (1-x)^2

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That's not true

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a^2 - b^2 =/= (a-b)^2

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It's easy to think that you can just do (a+b)^2 = a^2 + b^2

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But be careful! You can do that only if it was multiplication, not addition

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Like here, it's correct to say: (a * b)^2 = a^2 * b^2

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Becasue that's what ^2 means

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You just multiply the thing with itself

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(a * b)^2 = (a * b) * (a * b) = a * b * a * b = a * a * b * b = a^2 * b^2

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I belive you can notice now that

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(a+b)^2 = (a+b) * (a+b) = a * a + a * b + b * a + b * b = a^2 + 2 * a*b + b^2

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What all of this means that:

y^2 = 1 - x^2 ===> We got to square root if we want to write y as a function of x-es. So we get:

y = +- sqrt(1-x^2)

But this is a problem since functions can only output a single thing, not two! ( the function f(x) = +- sqrt(1-x^2) gives 2 things, the positive and the negative version of this thing). Functions don't that (by definition) and so we can't write y as a function of x. Using this argument again we can see that we can't write x as a function of y either.

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Your argument's aren't bad, it's just that a single thing at the beginning of your thinking can screw up everything, so don't feel too bad about this

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@smoky plover

smoky plover Sep 11, 2023, 4:41 PM

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amber pike Ahh ok I think I see that the problem is

well... that's the question given. I don't know if it's a circle of radius or not. That's why I need someone to explain it

amber pike Sep 11, 2023, 4:42 PM

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Yes it is

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That's what a circle is

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Points the same distance around a point

smoky plover Sep 11, 2023, 4:42 PM

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okay...

amber pike Sep 11, 2023, 4:42 PM

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If you have any other questions ask

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or if you didn't like my explanations or help just say so, someone else will help

smoky plover Sep 11, 2023, 4:43 PM

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so
If these are possible, then what will be the domains for these two functions? What are the graphs of these two functions?

smoky plover Sep 11, 2023, 4:43 PM

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amber pike If you have any other questions ask

no! I SINCERELY appreciate it

amber pike Sep 11, 2023, 4:44 PM

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They aren't possible in the first place so that question doesn't mean anything

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Or maybe I have misunderstood something

smoky plover Sep 11, 2023, 4:44 PM

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I am just yet to grasp what you're telling me

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so If You break it down a lil bit then it gonna be settled

smoky plover Sep 11, 2023, 4:51 PM

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amber pike Or maybe I have misunderstood something

However, a co-tenant of mine, said they are possible and I asked Chat GPT too, and the same response. I just need help with this:
If these are possible, what will be the domains for these two functions? What are the graphs of these two functions? What are the graphs of these two functions?
Are there points of the coordinate axes that relate to (0, 2) by means of R?

amber pike Sep 11, 2023, 5:00 PM

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I don't know since it depends on how the function looks

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Also I don't understand the last question either

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I think the answer to the last question would be "the coordonate points around (0,2) at 1 distance apart"

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Or the circle of radius 1 around (0,2)

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But it seems weird the professor to ask this question at the end since you need to know that this is circle and you couldn't have completed the first question otherwise

#can anyone cross-check this for me and help me explain the last one