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Which Relation Is Not a Function? Explanation and Examples

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.

X

$4$

$10$

$5$

$4$

$5$

Y

$8$

$20$

$16$

$30$

$35$

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:

X

$4$

$10$

$5$

$15$

$25$

Y

$8$

$20$

$16$

$30$

$35$

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:

Relations and functions

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 $y = 300 + x$.

X

$70$

$60$

$50$

$65$

$55$

Y

$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.

X

$-4$

$2$

$6$

$10$

$5$

Y

$2$

$-4$

$11$

$12$

$10$

2.

X

$-5$

$-10$

$10$

$15$

$20

Y

$5$

$15$

$5$

$14$

$35$

3.

X

$-3$

$0$

$5$

$7$

$11$

Y

$0$

$0$

$8$

$12$

$16$

4.       

X

$4$

$8$

$12$

$16$

$20$

Y

$6$

$12$

$18$

$24$

$30$

Solution:

  1. This is a function as each input has a unique output. No output is paired or mapped with two or more inputs.
  2. 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.
  3. 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.
  4. 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.

Relation pic

2.

Relation

3.

Relation

4.

Relation

Solution:

  1. This is not a function as two values of input are related to the same output value.
  2. This is a function as each value of the input is related to a single value of output.
  3. This is not a function as two values of input are related to the same output value.
  4. 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.

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

Solution:

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

Circle function

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.

X

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:

Sinx

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.

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