In mathematics, you will come across relations and functions quite often, but one burning question that arises in many students’ minds is which relation is not a function. A relation that does not have the properties of a function is just a simple relation. Every function is a relation but every relation is **not a function**.

*A relation in which every input has a single or unique output is termed a function.*

## Which Relation Is Not a Function?

A relationship between two or more variables where **a single or unique output does not exist for every input** will be termed a simple relation and not a function. In contrast, if a relationship exists in such a manner that there exists a single or unique output for every input, then such relation will be termed a function.

### Relation

A relation is defined as **the collection of ordered pairs from the given sets**. For example, if two sets A and B are given and we take an object “$*x$*” from set A and object “$*y$*” from set B, then both the objects are related to each other if they are put in ordered pair form (x, y). The relation is basically a relationship between input and output and it can be represented as (input, output).

Let us give an example to understand the concept of a relation. Anna has collected the data for two variables. The table represents **the data of the said variables**.

| $4$ | $10$ | $5$ | $4$ | $5$ |

| $8$ | $20$ | $16$ | $30$ | $35$ |

From the above table, we can see that for the input value of $4$ and $5$, we have **two outputs respectively**. Hence this set of ordered pairs is a relation and not a function.

Let us study now study an example of a relation that is a function as well.

*Anna collected data for two variables that are represented as:*

| $4$ | $10$ | $5$ | $15$ | $25$ |

| $8$ | $20$ | $16$ | $30$ | $35$ |

In this relation, each value of “$x$” **is related to a unique value of** “$y$”, hence it is a function.

### Function

A function is **a relation between two variables**. If two variables “$x$” and “$y$” are in a relation such that the change in the value of one variable results in a different value of the other variable, then we will say that the relation between two variables is a function. The function notation is given as $y = f(x)$. For every value of “$x$” there will be a unique value of “$y$”.

A relation between two sets A and B will be called a function, if **every element in set A has a single or unique image in set B**. In short, no two elements of set A can have two different images of Set B.

Hence, every relation is a function but not every function is a relation and *it can be represented as:*

You will not find which relation is not a function calculator online, so let us *study various examples* and numerical problems.

Anna is studying six subjects and her cumulative score is $300$ in five subjects. The final or total score will depend upon the marks obtained by Anna in mathematics. Assume “$x$” represents Ana’s marks in mathematics while “$y$” represents her cumulative score in six subjects. The relation between two variables can be written as $y = 300 + x$.

| $70$ | $60$ | $50$ | $65$ | $55$ |

| $300+70 = 370 | $300+60 = 360$ | $300+50 = 350$ | $300+65 = 365$ | $300 +55 = 355$ |

We can see that for every value of “$x$” we have a unique value of “$y$”. So in this case, we have **a unique output for every available input**. In the case of the function, all the available inputs are called the domain of the function and all the possible outputs are called the range of the function.

### Example 1:

The elements of the two set A and B are $A = {1 , 2, 3}$ to $B = {4, 5, 6}$. The relations formed by using above two sets are given as $X = {(1, 4), (3, 5)}$, $Y = {(1, 6), (1, 3), (3, 6)}$, $Z = {(1, 4), (2, 5), (3, 6)}$. You are required to determine or Identify which of these relations are functions.

### Solution:

Let us determine one by one whether the given relations are functions or not.

1) The first relation is $X = {(1, 4), (3, 5)}$. In this relation, two elements of set A are related to two elements of set B.

Hence, all the elements of set A are not mapped to elements of B which violates the condition of a relation to be a function. We have discussed that a function is a subset of relation, so it is bound to contain all elements of Set A and B. Hence, X** is not a function**.

2) The second relation is $Y = {(1, 6), (1, 3), (3, 6)}$. In this relation, two elements of set A are related to three elements of set B.

We can notice that the number “$1$” is paired with numbers “$6$” and “$3$”, hence one element in set A is mapped with two elements of set B and this violates the condition for a relationship to be a function. Hence, the relation Y** is not a function**.

3) The third relation is $Z = {(1, 4), (2, 5), (3, 6)}$. In this relation, all three elements of set A are related to all three elements of set B.

Furthermore, all the elements of set B are unique and there is no repetition or pairing of the same elements. Hence, relation Z **is a function**.

### Example 2:

The elements of the two set A and B are $A = {a , b, c, d}$ to $B = {v, x, y, z}$. The relations formed by using the two sets above are given as $X = {(a, v), (b, x), (c, z), (d, z)}$, $Y = {(a, v), (a, x), (a, y)}$, $Z = {(a, z), (b, x), (c, v), (d, y)}$. You are required to determine or Identify which of these relations are functions.

### Solution:

Let us determine one by one whether the given relations are functions or not.

1) The first relation is $X = {(a, v), (b, x), (c, z), (d, z)}$. In this relation, four elements of set A are mapped to three elements of set B.

We can notice that the element “z” is mapped twice with “c” and “d” respectively. Hence, all the elements of set A are not unique, so this relation has violated the condition of a function.

We can conclude that relation X **is not a function**.

2) The second relation is $Y = {(a, v), (b, x), (c, z), (d, z)}$. In this relation, only one element of set A is mapped to three elements of set B.

The letter “a” from set A is paired with letters “v”, “x”, and “y” from set B and it violates the condition of a function as one element cannot have multiple pairings. Hence, we can conclude the relation Y** is not a function**.

3) The third relation is $Z = {(a, z), (b, x), (c, v), (d, y)}$. In this relation, all four elements of set A are related to all the unique four elements of set B. As all the elements of set B are unique and repetition of elements is made in pairing.

Hence relation Z **satisfies the condition of a function**.

### Example 3:

For the set $X = {1, 3, 5, 7, 9, 11}$, define the relation from X to X in the form $R = {(x, y) : y = x + 2}$. Also determine the domain and range of R.

__Solution:__

The domain of a function is **the input values of the function**. In this relation, all the elements of set X are the domain of the function.

The domain of $R = {1, 3, 5, 7, 9, 11}$

*Let us now define the relation $R = {(x, y) : y = x + 2}$ in X to X form:*

- When $x = 1$, $y = 1 + 2 = 3$
- When $x = 3$, $y = 3 + 2 = 5$
- When $x = 5$, $y = 5 + 2 = 7$
- When $x = 7$, $y = 7 + 2 = 9$
- When $x = 9$, $y = 9 + 2 = 11$
- When $x = 11$, $y = 11 + 2 = 13$

All the values of “$y$” have images in “$X$” apart from $13$. Hence, **the range of function will be** $R = {3, 5, 7, 9, 11, 13}$.

### Example 4:

For the set $X = {1, 3, 5, 7, 9, 11}$, define the relation from X to X in the form $R = {(x, y) : y = x + 2}$. Also, determine the domain and range of R.

__Solution:__

The domain of a function is the input values of the function. In this relation, all the elements of set X are **the domain of the function**.

The domain of $R = {1, 3, 5, 7, 9, 11}$

*Let us now define the relation $R = {(x, y) : y = x + 2}$ in X to X form:*

- When $x = 1$, $y = 1 + 2 = 3$
- When $x = 3$, $y = 3 + 2 = 5$
- When $x = 5$, $y = 5 + 2 = 7$
- When $x = 7$, $y = 7 + 2 = 9$
- When $x = 9$, $y = 9 + 2 = 11$
- When $x = 11$, $y = 11 + 2 = 13$

All the values of “y” have images in “X” apart from 13. Hence, **the range of function will be** $R = {3, 5, 7, 9, 11, 13}$.

### Example 5:

From the data given below, determine which relation is a function.

1.

| $-4$ | $2$ | $6$ | $10$ | $5$ |

| $2$ | $-4$ | $11$ | $12$ | $10$ |

2.

| $-5$ | $-10$ | $10$ | $15$ | $20 |

| $5$ | $15$ | $5$ | $14$ | $35$ |

3.

| $-3$ | $0$ | $5$ | $7$ | $11$ |

| $0$ | $0$ | $8$ | $12$ | $16$ |

4.

| $4$ | $8$ | $12$ | $16$ | $20$ |

| $6$ | $12$ | $18$ | $24$ | $30$ |

__Solution:__

- This is a function as each input has a unique output. No output is paired or mapped with two or more inputs.
- This is not a function as the output value “$5$” is paired with input values “$-5$” and “10”, respectively, which violates the conditions of a function.
- This is not a function as the output value “$0$” is paired with input values “$-3$” and “0”, respectively, which violates the condition of a function.
- This is a function as each input has a unique output. No output is paired or mapped with two or more inputs.

### Example 6:

From the figures given below, find out which is not a function.

1.

2.

3.

4.

__Solution:__

- This is not a function as two values of input are related to the same output value.
- This is a function as each value of the input is related to a single value of output.
- This is not a function as two values of input are related to the same output value.
- This is a function as each value of the input is related to a single output. No input value has more than one output, hence it is a function.

## What Is Vertical Line Test of a Function/Relation?

The vertical line test is **a test used to determine whether a relation is a function or not**. To test the vertical line method, we need to first draw the graphical representation of the given equation/relation.

When the graph is drawn, we just draw a straight line with a pencil. If the line **touches the graph at two or more points**, then it is not a function; if the line touches the graph once, then the given equation or relation is a function.

### Example 7:

Draw the graph for the given equations/relations given below. You are also required to determine which of the given equations are functions by using the vertical line test.

- $x^{2}+ y^{2} = 3$
- $y = 3x + 5$
- $y = sin(x)^{2}$

__Solution:__

1. The equation* represents a circle* and the graph for the given equation is shown below.

As the straight line touches the graph at two points, hence the given equation/relation **is not a function**.

2. The equation or relation represents *a straight line* and its graph is shown below.

As the straight line touches the graph only once, hence** it is a function**.

3.The equation represent $sinx ^{2}$, *a trigonometric function*. Its graph *can be drawn as:*

As the straight line touches the graph only once, **it is a function**.

## Conclusion

After studying the in-depth comparison between a relation and a function, we can draw** the following conclusions**:

- Any relationship in which each input does not have a unique output is not a function.
- For a relation to be a function, the order pairing of the elements of the set or the mapping of the elements of sets should be unique, and each input should have a unique output for a relationship to be a function.
- To determine whether a graphical plot or drawing is a function or not, we can use a vertical line test. Draw a straight line and if it intersects the graph on more than one point, then the graph is not a function. If it crosses the graph only once, then the said graph is a function.

After reading this complete guide, we’re sure you now understand which relations are not functions.