Q.Which of the following relations are functions? Give reasons. If it is a function, determine its domain and range.

Concept: Relation Arrow Diagram — A relation is a function if every input (first element) maps to exactly one output (second element). No input can have two different outputs.

(i) $\{(2, 1), (5, 1), (8, 1), (11, 1), (14, 1), (17, 1)\}$

Each first element is unique — no repetition. So each input has exactly one output.

It is a function.

Domain = $\{2, 5, 8, 11, 14, 17\}$, Range = $\{1\}$.

(ii) $\{(2, 1), (4, 2), (6, 3), (8, 4), (10, 5), (12, 6), (14, 7)\}$

Again, every first element appears only once. Each input maps to a single output.

It is a function.

Domain = $\{2, 4, 6, 8, 10, 12, 14\}$, Range = $\{1, 2, 3, 4, 5, 6, 7\}$.

(iii) $\{(1, 3), (1, 5), (2, 5)\}$

Here, input $1$ maps to both $3$ and $5$ — two different outputs for the same input.

This violates the definition of a function.

It is not a function.

The core idea: the arrow diagram test

A relation is just a set of ordered pairs. To decide if it’s a function, think of each first element as an input and each second element as its output. The rule is simple: one input cannot point to two different outputs. If you draw arrows from inputs to outputs, no input should have more than one arrow leaving it.

This is the vertical line test in disguise — but for discrete points, the arrow diagram is clearer.

(i) $\{(2, 1), (5, 1), (8, 1), (11, 1), (14, 1), (17, 1)\}$

1. Check the inputs: The first elements are $2, 5, 8, 11, 14, 17$. Each appears exactly once.

   No input is repeated, so there’s no chance of one input having two different outputs.

   That alone satisfies the function condition.

2. Domain: The set of all first elements — $\{2, 5, 8, 11, 14, 17\}$.

3. Range: The set of all second elements — here every pair has $1$ as the output.

   So the range is just $\{1\}$.

(ii) $\{(2, 1), (4, 2), (6, 3), (8, 4), (10, 5), (12, 6), (14, 7)\}$

1. Check the inputs: First elements are $2, 4, 6, 8, 10, 12, 14$ — all distinct.

   No input repeats, so it’s a function.

2. Domain: $\{2, 4, 6, 8, 10, 12, 14\}$.

3. Range: The second elements are $1, 2, 3, 4, 5, 6, 7$ — all distinct as well.

   So the range is $\{1, 2, 3, 4, 5, 6, 7\}$.

(iii) $\{(1, 3), (1, 5), (2, 5)\}$

1. Check the inputs: The input $1$ appears twice — once with output $3$ and once with output $5$.

   That means the same input $1$ gives two different outputs.

   This violates the definition of a function.

2. Conclusion: It is not a function.