Does this equation determine a relation between x and y?

Question

Consider x² + (y-2)² = 1 and the relation "(x,y) R(0,2)", where R is read as"has distance 1 of".

For example, "(0,3) R(0,2)". This relation can also be read as "the point (x,y) is on the circle of radius 1 with centre (0,2)". In other words: "(x,y) satisfies this equation x² + (y - 2)² = 1, if and only if (x,y) R(0,2)".

Does this equation determine a relation between x and y? Can the variable x 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?

Answer 1

x^2 + (y-2)^2 = 1 is a circle with radius=1 and center (0,2)

you can solve it for either x or y, but they aren't functions, just relations, since a circle is not a function, as you can draw vertical lines that cut the circle in more than one point.

You could try to split the relation in two, and make 2 functions that are equivalent to the circle, 2 semi-circles

x = +sqr(1- (y-2)^2) and

x =-sqr(1-(y-2)^2)

x=f(y) and x=g(y) are 2 semi-circles, but not functions, just relations. they are the left half of a circle and the right half

and

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

y-2 = sqr(1-x^2)

y = 2+sqr(1-x^2) or y = j(x)

and y= 2-sqr(1-x^2) or y=k(x)

j(x) and k(x) are semicircles