Algebra Question

Question

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 “(x, y) belongs to 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, and y be seen as a function of x? What are 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

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

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

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

the domain is 1-x^2 >=0

1 >= x^2

[-1, 1] is the domain

the inverse is:

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

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

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

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

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

+or-sqrt( 1 - x^2 + 4x - 4) = y

+or-sqrt( -3 + 4x - x^2) = y

+or-sqrt( - (x^2 - 4x +3)) = y

+or-sqrt( - ( x - 3)(x - 1 )) = y

the domain of the inverse is :

-(x-3)(x-1) >= 0

(x-3)(x-1) <= 0

the domain is [1,3]