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Into Function
Into function is a type of function where at least one element of the codomain will not have a preimage in the domain. Suppose there are two sets, A (domain) and B (codomain). If at least one element of set B is not associated with an element in set A then such a function will be known as an into function.
The range of an into function will be a subset of the codomain. However, the range will not be equal to the codomain. The elements of an into function are usually represented as an ordered pair of the form (input, output). In this article, we will learn more about an into function, its meaning, as well as associated examples.
1.  What is Into Function? 
2.  Into Function Graph 
3.  Into Function vs Onto Function 
5.  FAQs on Into Function 
What is Into Function?
An into function is a type of function that establishes a binary relation between two sets such that every element of the first set (domain) will be associated with exactly one element of the second set (codomain) and at least one element of the codomain will not be associated with any element in the domain.
Into Function Definition
For a function f: A \(\rightarrow\) B to be an into function there will be one or more elements in set B that do not have a preimage in set A. In other words, in an into function, not all elements of the codomain will be mapped to the elements of the domain. As a consequence, the range of an into function will be a subset of the codomain however, the range and codomain will never be equal.
Into Function Example
Let set P = {1, 2, 3} and set Q = {7, 8, 9, 10} be defined by the function f = {(1, 7), (2, 9), (3, 8)}. As element 10 of set Q does not have a preimage in set P thus, this function is an into function. The mapping of an into function can be done with the help of an arrow diagram given as follows:
Into Function Graph
To check whether a graph represents an into function or not, the vertical line test will be used. The steps to perform the vertical line test can be understood using an example. Suppose the function is given as f = {x, 1 < x < 1}. Then the verticle line test can be conducted as follows:
 Draw a vertical line at any arbitrary point on the xaxis. Say a line is drawn at x = 0.5.
 Determine the number of points where the vertical line intersects with the graph. If the vertical line meets the graph at exactly one point then the graph is a function. This signifies that the x value will have exactly one output thus, by definition, the graph will be a function. If the vertical line intersects the graph at more than one point then it means that the x value will have more than one output hence, the graph will not be a function.
 In the example, y = x, as the vertical line will intersect the graph at exactly one point, hence, it is a function.
Into Function vs Onto Function
Into functions and onto functions help to establish a relationship between the elements of the domain and codomain of two given sets. The table below lists the main differences between an into function and onto function.
Into Function  Onto Function 
In an into function, there will be at least one element in the codomain that does not have a preimage in the domain.  In an onto function, every element in the codomain will have at least one preimage in the domain. It is also known as subjective mapping. 
Example: P = {a, b, c}, Q = {1, 2, 3, 4} and f = {(a, 1), (b, 2), (c, 3)}  Example: P = {1, 2, 3, 4}, Q = {a, b, c} and f = {(1, b), (2, a), (3, c), (4, c)} 
The arrow diagram for an into function is given as follows: 
The arrow diagram for an onto function is given below: 
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Important Notes on Into Function
 An into function is a type of function where one or more elements of the codomain will not have a preimage in the domain.
 The range of an into function will be a subset of the codomain. The range will not be equal to the codomain.
 The vertical line test is used to check if a given graph is a function or not. If the graph intersects with the vertical line at exactly one point, it will be a function.
Examples on Into Function

Example 1: Let A = {1, 4, 7, 12}, B = {a, b, c, d, e, f} and f = {(1, a), (4, f), (7, c), (12, d)}. Is the function an into function?
Solution: Only the elements, a, c, d, and f, of the codomain B, have a preimage in domain A.
This implies that b and e are not mapped to any element in A.
Thus, by definition, the given function is an into function.
Answer: The function is an into function

Example 2: Does the given mapping represent an into function?
Solution: Domain of the function, A = {1, 2, 3, 4}Codomain of the function, B = {a, b, c, d}
f = {(1, d), (2, a), (3, c), (4, b)}
As every element of the codomain has a preimage in the domain, thus, the given mapping is not an into function.
Answer: The given map does not represent an into function.

Example 3: Will x^{2} + y^{2} = 9 represent an into function?
Solution: First, we check if the given equation represents a function. The graph of x^{2} + y^{2} = 9 will be a circle that has its center at the origin and the radius will be 3.
We draw the graph and then use the vertical line test to check if the given equation is a function as follows:
As the vertical line cuts the graph at two points, thus, the given equation is not a function. Hence, as a consequence, it will not be an into function.Answer: x^{2} + y^{2} = 9 is not a function hence, it is not an into function.
FAQs on Into Function
What Does Into Function Mean in Math?
Into function refers to the binary relation between two sets where at least one element of the codomain will not be mapped to an element in the domain.
What is Into Function Also Called?
An into function does not have an alternative name. However, onto functions are known as surjective functions, onetoone are injective functions, and functions that are both onto and onetoone are bijective functions.
What is an Example of Into Function?
Suppose set X = {1, 2, 3} and set Y = {10, 20, 30,40}. If a function is defined as f = {(1, 10), {2, 20}, (3, 30)} then f is an into function. This is because 40, from set Y, is not mapped to any element in set X.
What is the Domain of Into Function?
The complete set of values of the independent quantity or the input of an into function is known as the domain of that function.
What is the Range of Into Function?
The range of an into function are all the output values that result from a given input. The range of an into function will be a subset of the codomain but will never be equal to it.
What is Onto and Into Function?
An onto function is one in which all elements of the codomain are mapped to one or more elements in the domain. In an into function, there will be at least one element in the codomain that will not be mapped to any element in the domain.
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