Question

Which of the following relations are functions? Give reasons. If it is a function, determine its domain and range.
$\left(i\right)\left\{\right(2,1),(5,1),(8,1),(11,1\left)\right(14,1),(17,1\left)\right\}$
$\left(ii\right)\left\{\right(2,1),(4,2),(6,3),(8,4\left)\right(10,5),(12,6),(14,7\left)\right\}$
$\left(iii\right)\left\{\right(1,3),(1,5),(2,5\left)\right\}$

Answer 1

$\left(i\right)$ Given:$\left\{\right(2,1),(5,1),(8,1),(11,1),(14,1),(17,1\left)\right\}$
Step 1:Checking for function
Since, every element has one and only one image.
So, the relation is a function.

Step $2:$ Domain of the function
Domain $=$ set of first elements of ordered pairs in the given relation
$=\{2,5,8,11,14,17\}$

Step $3:$ Range of the function
Range $=$ set of second elements of ordered pairs in the given relation
$=\left\{1\right\}$
Hence, given relation is a function whose domain is $\{2,5,8,11,14,17\}$ and range is $\left\{1\right\}$.

$\left(ii\right)$ Given:$\left\{\right(2,1),(4,2),(6,3),(8,4\left)\right(10,5),(12,6),(14,7\left)\right\}$
Step $1:$ Checking for function
Since, every element has one and only one image
So, the relation is a function.

Step $2:$ Domain of the function
Domain $=$ set of first elements of ordered pairs in the given relation
$=\{2,4,6,8,10,12,14\}$

Step $3:$ Range of the function
Range $=$ Set of second elements of ordered pairs in the given relation $=\{1,2,3,4,5,6,7\}$
Hence,given relation is a function whose domain is $\{2,4,6,8,10,12,14\}$and range is $\{1,2,3,4,5,6,7\}$.

$\left(iii\right)$ Given:$\left\{\right(1,3),(1,5),(2,5\left)\right\}$
As we can see ,first element i.e.,$1$ corresponds to two different images i.e.,$3$ and $5,$ this relation is not a function.
Hence, given relation is not a function