Question

Consider the relation between two sets $X$ and $Y,$ where $x$ belongs to $X$ and $y$ belongs to $Y.$ Which of the following relations between $X$ and $Y$ is a function?
I. $x-y=2$
II. $x^2+y^2=8$
III. $\left\{\right(2,3),(2,4\left)\right\}$
IV. $\left\{\right(2,4),(3,4\left)\right\}$

• A. Both II and IV.
• B. Both I and IV.
• C. Both I and II.
• D. Both II and III.

Answer 1

The correct option is B Both I and IV.

A function is a relationship between two sets in which each element of the input set is associated with exactly one element of the output set.

Let’s check all the options one by one.

Statement I:
$x-y=2$
$y=x-2$
For every value of x, the value of y will be $2$ less than the value of $x.$ This means we cannot have the same value of y for different values of $x.$
Hence, $x-y=2$ is a function.

Statement II:
$x^2+y^2=8$
$\Rightarrow y^2=8-x^2$
$\Rightarrow y=\sqrt{8-x^2}$
When $x=2,y$ can have the values $2$ as well as $-2.$ This is because when every number has two square roots, i.e., the principal square root, and its negation. So, for one value of $x$, there is more than one value of $y.$
Hence, $x^2+y^2=8$ is not a function.

Statement III:
$\left\{\right(2,3),(2,4\left)\right\}$
When $x=2,y$ has two values, $3$ and $4$
Hence, $\left\{\right(2,3),(2,4\left)\right\}$ is not a functon.

Statement IV:
$\left\{\right(2,4),(3,4\left)\right\}$
Both inputs $(x=2,3)$, are related to one output, i.e., $4.$
Hence, $\left\{\right(2,4),(3,4\left)\right\}$ is a function.