# Which of the following relations are functions? Give reasons

Question:

Which of the following relations are functions? Give reasons. In case of a function, find its domain and range

(i) $f=\{(-1,2),(1,8),(2,11),(3,14)\}$

(ii) $g=\{(1,1),(1,-1),(4,2),(9,3),(16,4)\}$

(iii) $h=\{(a, b),(b, c),(c, b),(d, c)\}$

Solution:

For a relation to be a function each element of $1^{\text {st }}$ set should have different image in the second set(Range)

i) (i) $f=\{(-1,2),(1,8),(2,11),(3,14)\}$

Here, each of the first set element has different image in second set.

$\therefore f$ is a function whose domain $=\{-1,1,2,3\}$ and range $(f)=\{2,8,11,14\}$

(ii) $g=\{(1,1),(1,-1),(4,2),(9,3),(16,4)\}$

Here, some of the first set element has same image in second set.

$\therefore \mathrm{g}$ is not a function.

(iii) $h=\{(a, b),(b, c),(c, b),(d, c)\}$

Here, each of the first set element has different image in second set.

$\therefore \mathrm{h}$ is a function whose domain $=\{\mathrm{a}, \mathrm{b}, \mathrm{c}, \mathrm{d}\}$ and range $(\mathrm{h})=\{\mathrm{b}, \mathrm{c}\}$

(range is the intersection set of the elements of the second set elements.)