Mappings of Functions (Two cases)

For a mapping to be a function, we can have two cases. The first occurs when the mapping is one-to-one and the second case occurs when the mapping is many-to-one. Essentially, this means that a function can have multiple input values producing the same output value, but not multiple outputs produced by a single input.

Here, we will learn about the two cases of mapping that are considered functions. In addition, we will look at some examples to apply the concepts.

ALGEBRA
Mapping of a one-to-one function

Relevant for

Learning about the mapping of functions.

See mappings

ALGEBRA
Mapping of a one-to-one function

Relevant for

Learning about the mapping of functions.

See mappings

Case (i): One-to-one mapping

We are going to consider two non-empty sets A and B. A mapping from A to B is a rule that associates each element of set A with an element of set B.

A mapping can be represented using a mapping diagram. Consider the following mapping that has the sets A={-2, -1, 0, 1, 2} and B={0, 1, 2, 3, 4, 5, 6}.

Mapping of a one-to-one function

In case (i), we observe that each element of set A is related to only one element of set B. This is a so-called one-to-one mapping.

The fact that there are no elements of set A that are related to elements 0 or 1 in set B is not important.

A one-to-one mapping is called a function. Usually, we denote using f the rule that associates each element of A with an element of B.

For example, in the mapping above, the rule is “add 4”. Using function notation, we can write:

$latex f(x)=x+4~~$ or $latex ~~f:~x\rightarrow x+4$


Case (ii): Many-to-one mapping

Again, we are going to consider the sets A={-2, -1, 0, 1, 2} and B={0, 1, 2, 3, 4, 5, 6}, but in this case, we have the following mapping diagram:

Mapping of a many-to-one function

In case (ii) we observe that two elements of set A are related to one element of set B. This is either a two-to-one mapping or a many-to-one mapping.

A many-to-one mapping is also called a function.

For example, in the mapping above, the rule is “square”. Using function notation, we can write:

$latex f(x)=x^2~~$ or $latex ~~f:~x\rightarrow x^2$


Examples of mappings that are functions

In the following examples, we can apply what we have learned about one-to-one and many-to-one mapping to determine what type of relationship the following functions are.

EXAMPLE 1

Determine whether the function $latex f(x)=\frac{x}{2}+2$ is a one-to-one or many-to-one function.

Solution: The graph of the function f is as follows

Graph of a one-to-one function

We can see that the graph of $latex f(x)=\frac{x}{2}+2$ is a straight line. Furthermore, we clearly see that the function is one-to-one since each value of x is related to only one value of y.

EXAMPLE 2

Determine whether the function $latex f(x)=x^2+1$ is a one-to-one or many-to-one function.

Solution: Since we have $latex f(-1)=(-1)^2+1=2$ and $latex f(1)=1^2+1=2$, we know that the function is not one to one. Its graph is:

Graph of a many-to-one function

Looking at the graph of the function, we see that this function is many to one, specifically two to one.

EXAMPLE 3

Is the function $latex f(x)=(x^4+1)^2-3$ a one-to-one or many-to-one function?

Solution: One way to determine if a function is many-to-one is to use the $latex f(-1)=f(1)$ test. Therefore, we have:

$latex f(1)=(1^4+1)^2-3$

$latex =(2)^2-3$

$latex =4-3$

$latex =1$

$latex f(-1)=((-1)^4+1)^2-3$

$latex =(2)^2-3$

$latex =4-3$

$latex =1$

We see that $latex f(-1)=f(1)$ is true. This means that the function is many to one.

Note: Consider that the $latex f(-1)=f(1)$ test works only when the function is symmetric about the y-axis.


See also

Interested in learning more about algebraic functions? Take a look at these pages:

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