Let `A={x in R:- 1 <= x <= 1}=B and c={x in R: x >= 0}and` let `S={(x,y)in A xx B: x^2+y^2=1} and S_0={(x,y)in A xx C: x^2+y^2=1}.` The

Options

A. S defines a function from A to B
B. `S_(0)` defines a function from A to C
C. `S_(0)` defines a function from A to b
D. S defines a function from A to c

Answer

B

Explanation

We observe that S denotes the set of all points on the circle `x^(2)+y^(2)=1 and S_(0)` denotes the set of all points on the semi-circle `x^(2)+y^(2)=1` lying above x-axis. Clearly ,`(1//2, sqrt3//2),(1//2,-sqrt3//2) in S`. Infact corresponding to every value of `x in A` there are two values of `y in B` Therefore, S is not a function from A to B. Clearly `S_(0) subset AxxC` such that for each `x in A` there is unique value of `y in C`. Therefore, `S_(0)` is a function from A to C.
ALTER
If a straight line parallel to y-axis cuts the graph of a curve at more than one-point them the set of all points on it does not define a function. However, the set of all points on a curve defines a function when no line parallel to y-axis crosses it more than once.