Question

Consider relations between two sets $X$ and $Y$ where $x$ belongs to $X$ and $y$ belongs $Y$, find which of the following relation from $X$ to $Y$ is a function.

• A. $x-y=2$
• B. $x^2+y^2=8$
• C. $\left\{\right(2,3),(2,4\left)\right\}$
• D. All of the above.

Answer 1

The correct option is A $x-y=2$

We know, function is a relation between two sets in which each element of input set is associated with an exacly one element of the output set.

Let us check all the options one by one

$\underset{–}{Option\left(a\right)}$,
$x-y=2\Rightarrow y=x-2$
now, for every value of $x$, the value of $y$ will be $2$ less than the value of $x$ that means we cannot have the same value of $y$ for different values of $x$.
Hence, $x-y=2$ is Function.


$\underset{–}{Option\left(b\right)}$,
$x^2+y^2=8$
when $x=2$, $y$ can have the values $2$ as well as $-2$. As for one value of $x$, there are more than one values of $y$ hence $x^2+y^2=8$ is NOT a function.

$\underset{–}{Option\left(c\right)}$,
$\left\{\right(2,3),(2,4\left)\right\}$
when $x=2$, $y$ has two values as $3$ and $4$. So it is also not Function

Hence option(a) is the only correct answer here.