Prove if equation if equation determines relation between x and y. College education.

[panda_user]

Consider the equation [imath]x^2 + (y - 2)^2 = 1[/imath] 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 [imath]x^2 + (y - 2)^2 = 1[/imath], if and only if, (x, y) R (0, 2)”.

1) Does this equation determine a relation between x and y?
2) Can the variable x can be seen as a function of y, like x=g(y)?
3) Can the variable y be expressed as a function of x, like y= h(x)?
4) If these are possible, then what will be the domains for these two functions?
5) What are the graphs of these two functions?
6) Are there points of the coordinate axes that relate to (0, 2) by means of R?

This is the assignment. I struggle to understand what is the first step to do here. How to determine the relation? And how to apply answers to all the other questions of the assignment? I just need a human-like simple explanation of what to even do here. Because algebra is one of the subjects I have a problem with. Thank you in advance to any help.

 

[stapel]

(FYI for other users: The original post had inline images and a paragraph of questions. Post has been edited to have inline math and numbered questions.)

Consider the equation [imath]x^2 + (y - 2)^2 = 1[/imath] 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 [imath]x^2 + (y - 2)^2 = 1[/imath], if and only if, (x, y) R (0, 2)”.

1) Does this equation determine a relation between x and y?
2) Can the variable x can be seen as a function of y, like x=g(y)?
3) Can the variable y be expressed as a function of x, like y= h(x)?
4) If these are possible, then what will be the domains for these two functions?
5) What are the graphs of these two functions?
6) Are there points of the coordinate axes that relate to (0, 2) by means of R?

This is the assignment. I struggle to understand what is the first step to do here. How to determine the relation? And how to apply answers to all the other questions of the assignment? I just need a human-like simple explanation of what to even do here. Because algebra is one of the subjects I have a problem with. Thank you in advance to any help.

1) What is your book's (or class') definition of "relation"? Given an [imath]x[/imath]-value, does the equation let you find the corresponding [imath]y[/imath]-value(s)?

2) How does your textbook / class notes define "function"?

3) (Same question as (2).)

4) This gives away the (expected) answers to (2) and (3) (though I would strongly disagree).

5) What is a "circle"? What does its graph look like?

6) Points on the axes are the intercepts.

 

[The Highlander]

Consider the equation [imath]x^2 + (y - 2)^2 = 1[/imath] 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 [imath]x^2 + (y - 2)^2 = 1[/imath], if and only if, (x, y) R (0, 2)”.

1) Does this equation determine a relation between x and y?
2) Can the variable x can be seen as a function of y, like x=g(y)?
3) Can the variable y be expressed as a function of x, like y= h(x)?
4) If these are possible, then what will be the domains for these two functions?
5) What are the graphs of these two functions?
6) Are there points of the coordinate axes that relate to (0, 2) by means of R?

This is the assignment. I struggle to understand what is the first step to do here. How to determine the relation? And how to apply answers to all the other questions of the assignment? I just need a human-like simple explanation of what to even do here. Because algebra is one of the subjects I have a problem with. Thank you in advance to any help.

It's a simple enough matter to see what a graph of this equation (or, indeed, any other equation) looks like by using the desmos (online graphing calculator); that may help you to answer some of your other questions too. ?

(Try re-arranging your equation and see what happens on there; just type "sqrt" to enter a radicand in desmos.)

 

[stapel]

...type "sqrt" to enter a radicand in desmos.

My issue is with the "[imath]\pm[/imath]" part of solving for one of the variables.

 

[The Highlander]

My issue is with the "[imath]\pm[/imath]" part of solving for one of the variables.

If you do use "±" and set it equal to [1, -1] in the slider, then that "works" fine but then "g(y)" & "h(x)" are no longer "functions" so I suspect the OP is expected to recognize that fact (and not use "±") so that s/he can correctly answer part "4)" about the "domains" of the "functions" that might be 'created'. (Same applies for attempting part "5)".)

NB: (Mainly for Steven's benefit because I didn't get around to answering him the last time he asked me about it.) When I put something in inverted commas (in a different typeface) then that is because I am directly quoting what someone else has written.