Which of the Following are Functions?
Which of the following are functions?
(a) \( \{(x,y): y^2=x,\; x,y\in R\} \)
(b) \( \{(x,y): y=|x|,\; x,y\in R\} \)
(c) \( \{(x,y): x^2+y^2=1,\; x,y\in R\} \)
(d) \( \{(x,y): x^2-y^2=1,\; x,y\in R\} \)
A relation is a function if each value of \(x\) gives exactly one value of \(y\).
In option (a),
\[ y^2=x \Rightarrow y=\pm\sqrt{x} \]
One value of \(x\) gives two values of \(y\).
So, not a function.
In option (b),
\[ y=|x| \]
Every value of \(x\) gives exactly one value of \(y\).
So, it is a function.
In option (c),
\[ x^2+y^2=1 \Rightarrow y=\pm\sqrt{1-x^2} \]
Not a function.
In option (d),
\[ x^2-y^2=1 \Rightarrow y=\pm\sqrt{x^2-1} \]
Not a function.
Hence,
\[ \boxed{\text{Correct Answer: (b)}} \]