112 LearnersLast updated on July 9th, 2025
Into Function
In mathematics, a function is a rule that allocates exactly one output value to each input value. The output value is assigned from the range, and the input value from the domain. This article talks more about functions, their properties, and graphs.
What is Into Function?
In mathematics, some functions are used to show the connection between two sets. One such function is an into function, where at least one element in the codomain is not the image of any element from the domain. Let’s consider an example where set A and set B are the domain and codomain respectively. Here, at least one element in set B is not the image of any element in set A. It can be denoted as f: A → B.
In an into function, the range is a proper subset of the codomain. Unlike an into function, an onto function maps every element in the codomain to at least one element in the domain.
Difference Between Into and Onto Function
There are two types of functions: into and onto. This section explains the difference between them.
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Into Function |
Onto Function |
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In into function, the codomain will have at least one element not in the image of any element from the domain. |
In an onto function, all the elements in the codomain are in the image of at least one element from the domain. So, it is surjective |
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At least one element in the codomain is not in the range. |
In an onto function, all the elements are in the range. |
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In a visual representation, at least one element in the codomain has no arrow pointing to it from the domain |
In a visual representation, each element in the codomain has at least one arrow pointing to it from the domain |
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The range, which is a proper subset of the codomain, is different from the codomain itself. |
Every element in the codomain is in the image of at least one element from the domain. |
What are the Properties of an Into Function?
The properties of an into function helps solve problems involving into functions. Here are some properties of into function.
- In an into function, all the elements in the domain are mapped to at least one element in the codomain. However, not every element in the codomain is the image of an element from the domain.
- At least one element in the codomain does not have a pre image in the domain.
- In an into function, the range is a subset of the codomain, and does not include every element of the codomain.
- An into function may be injective, but the terms are not the same. A function is called injective when each element in the domain maps to a distinct element in the codomain. On the other hand, an into function is a function where not all elements in the codomain are mapped to the domain.
Into Function Graph
To check if the graph represents an into function, the vertical line test is used. This test is used for all functions. Let’s understand this using simple examples.
First, we draw a vertical line at any random point on the x-axis, for example, x = 0.5.
Determine the number of times the vertical line intersects the graph.
If it intersects only once, the graph represents a function where each value of x corresponds to exactly one output.
If the vertical line intersects the graph at more than one point, it does not represent a function, because one input has multiple outputs.
For the function: f(x) = x2 for -2 < x < 2
Draw a vertical line at any value of x. For example,
x = 1.5
x = 0
x = -1
The function intersects the codomain at only one point, meaning not all codomain elements are mapped, so it is a function. The same test can be used to confirm an into function.
Real-world Applications of Into Function
When learning the into function, students should understand how to apply this function in real life. Here are some of the applications of into function.,
- We use into function to record the highest temperature each day. If days of the week are the domain, then all possible temperatures are the codomain.
- In sports, to assign players specific positions in games, we use the into function.
- Each student has a unique ID and is registered for a specific set of courses. But the list of all courses offered (codomain) may include some that no student chose. This is an example of the application of an into function in real life.
Common Mistakes and How to Avoid Them in Into Function
Students often make mistakes when solving problems involving into functions. These mistakes can be avoided if we are aware of the properties of into function and practice regularly. The below mentioned common mistakes will help us avoid them while dealing with into function.
Mistake 1
Confusing into with onto function
Students confuse into functions with onto functions. To avoid this confusion, always remember that in into function, the range is a subset of the codomain, but they are not the same. When it comes to the onto function, the range equals the codomain.
Mistake 2
Choosing an incorrect codomain
Students must choose the codomain carefully so that it aligns with the function. Failing to do so will lead to mistakes. To avoid this, always select a codomain carefully, so that it includes all possible outputs of the function, and always check if the codomain is larger than the range or not. Because in an into function, the codomain is larger than the range.
Mistake 3
Misinterpreting f: A → B
Students misinterpret f: A → B means the function maps all elements in the codomain B, which is wrong, and conclude the function is onto. To avoid this confusion, always understand that the notation, here f: A → B, tells us that A is the domain and B is the codomain, but does not state that every element in B is mapped. Also, identify the range and compare it with the codomain.
Mistake 4
Confusing domain and codomain
Confusion between domain and co-domain leads to incorrect calculation of function, especially when determining the function.
Mistake 5
Misreading the function’s graph
Students often misread the graph of an into function. To avoid this, always check the range by examining the y-values that the graph covers. Use the vertical line test to verify if the graph represents a function that is onto or into.
Solved Examples of Into Function
Problem 1
A function f maps the set {1, 2, 3} to {a, b, c, d}, where: f(1) = a, f(2) = b, f(3) = c. Is this function an into function?
Yes, the function is an into function
Explanation
The codomain of the function is {a, b, c, d}; it has 4 elements
The domain of the function is {1, 2, 3}, it has 3 elements
The function is into function.
Problem 2
A function f maps the set {1, 2, 3, 4} to {a, b, c} as follows: f(1) = a, f(2) = a, f(3) = b, f(4) = c. Is this function an into function?
The function is not an into function
Explanation
The elements in the codomain are {a, b, c}
Here, every element in the codomain is mapped to the domain, so this is onto, not into.
Problem 3
Is the function given into function. Let f : R → R be defined as f(x) = x2 + 1
Yes, the function is into
Explanation
the given function,
f(x) x2 + 1, the output is x2 + 1 is always ≥ 1, because the smallest value x2 can take is 0
Therefore, f(x) = x2 + 1 ≥ 1 for all x
So the range is [1, ∞)
Thus, the function is an into function
Problem 4
Let A = {2, 4, 6} and B = {1, 2, 3, 4, 5, 6}. Define f(x) = x for x ∈ A. Check if the function is an into function?
The function is into function
Explanation
Given,
Codomain function = {1, 2, 3, 4, 5, 6}
Elements in domain = {2, 4, 6}
The elements 1, 3, and 5 are not in the range.
So, the function is into.
Problem 5
Let f: R → R be defined by f(x) = ex. Is this function an into function?
The function is an into function
Explanation
The function is: f(x) = ex
Where the domain is R and the codomain is R
The exponential function f(x) = ex always gives a positive result:
Range of f(x) = (0, ∞)
Examples: f(0) = e0 = 1
f(1) = e1 = 2.7181 = 2.718
f(-1) = e-1 = 2.718-1 = 0.367
This shows that f(x) never gives negative values.
So the range is only positive real numbers.
Therefore, the function is into.
FAQs on Into Function
1.What is an into function?
2.What is the difference between an into function and an onto function?
3.What is the range of an into function?
4.How to check if a function is into?
5.What is the domain of an into function?
6.How does learning Algebra help students in Philippines make better decisions in daily life?
7.How can cultural or local activities in Philippines support learning Algebra topics such as Into Function ?
8.How do technology and digital tools in Philippines support learning Algebra and Into Function ?
9.Does learning Algebra support future career opportunities for students in Philippines?
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Jaskaran Singh Saluja
About the Author
Jaskaran Singh Saluja is a math wizard with nearly three years of experience as a math teacher. His expertise is in algebra, so he can make algebra classes interesting by turning tricky equations into simple puzzles.
Fun Fact
: He loves to play the quiz with kids through algebra to make kids love it.
