Into Functions

Into Functions study material

The function satisfies the condition of having no preimage in the domain for at least one element of the codomain. Let us consider two types of sets — A and B — where A contains the domain and B contains the codomain. The function will be an Into function if A has no preimage in the domain for at least one element of the B.

You will find that the range of the Into function will be the subset of the codomain. But it’s not always possible that the range will be equal to the codomain. You will get to know more about this topic with this article.

What is a Function?

In mathematics, a function is a mathematical expression that takes something as an input and gives output after proper calculation. To define a function in mathematics, we use the term f(x). Some examples of function are f(x)= [x], f(x)= |x|, or f(x)= x2. 

Example of a Function

Let us consider one example of the function f(x)= x2. This is an example of a squaring function. 

• Here x is the input variable.
• f denotes the function.
• x2 depicts that the function receives x as an input and then squares it as the output. 

What is an Into Function?

You could explain the Into function to establish a binary relation among two sets, A and B so that each element of the set A will be related to at least one element of set B, i.e., codomain. And at least one element of set B will not be related to any element of set A. This type of function is known as an Into function.  

Definition of Into Function

The mathematical definition of an Into function is as follows:

If a function f: A to B needs to be an “Into function”, there will be at least one or more elements in set B that will not have a preimage in the set A. All the elements of the codomain don’t need to be mapped with the elements of the domain. So, you can conclude that the range of the Into function is the subset of the codomain, but it’s not necessary that the range will be equal to the codomain. 

Example of an Into Function

Let us consider two sets — A and B — where set A={1,2,3} and B={7,8,9,10}. If the sets are to be defined in a function f={(1, 7), (2, 9), (3, 8)}. With the function f, you will observe that element 10 of set B is not having a preimage in set A. The range of the function, i.e., {7,9,8} is not equal to the codomain, i.e, {7,8,9,10}. So, this is an example of an Into function. 

Test of an Into Function

One can check whether a graph represents an Into function by drawing a vertical line. The test is known as the vertical line test. Whenever you are provided with a function, draw the graph of that function on the graph sheet. Draw a straight line such that it cuts the x-axis. Now observe the following things. 

•  If the vertical line crosses the graph at more than one point, it means it’s not an Into function. 
• However, if the line cuts the graph at only one point, then it’s an Into function. 

Example of a Vertical Line Test

To verify that a function is an Into function, you need to conduct a vertical line test. Let us understand it with the help of an example of a function. There is a function f= {|x|, -1 < x < 1}. This depicts a modulus function x. To conduct a vertical line test on this function, you must follow this-

• Draw the graph of modulus function on a graph sheet.
• Now, draw a vertical line on any point cutting the x-axis. For example, you draw a vertical line at x= 0.5. 
• Observe the number of points where the vertical line cuts the graph of the function. 
• As per the graph, there will be two observations.

1. If the vertical line is drawn, and it intersects the graph at only one point, then the graph is a function.
2. If the vertical line intersects the graph at more than one point, then the graph is not a function. 

The vertical line in this function will intersect at only one point. Hence, it is an Into function. 

Into Vs Onto Function

You might find it difficult to distinguish between an Into and an Onto function. However, keep these points in mind. 

• In the case of an Into function, at least one element in the codomain will not have a preimage in the domain. While in that of an Onto function, every codomain element has a preimage in the domain. 
• In the Into function, the range is not equal to the codomain. While in the Onto function, the range is always equal to the codomain. 

Conclusion

In mathematics, a function is a mathematical expression that takes something as an input and gives output after proper calculation.One example of the function f(x)= x2.If a function f: A to B needs to be an “Into function”, there will be at least one or more elements in set B that will not have a preimage in the set A.One can check whether a graph represents an Into function by drawing a vertical line.