Into Function - Definition, Graph, Examples

What is Into Function?

In the study of set theory and mappings, functions are categorised based on how elements of the first set (domain) relate to the elements of the second set (codomain). To understand this, we must look at the "completeness" of the mapping.

A function is considered an "into" type if there is "room to spare" in the codomain. Imagine two sets, A and B. If you map every element from A to B, but find that there is at least one element in set B that does not have a partner in set A, you have successfully created an into mapping. In technical terms, the range (the set of actual outputs) is smaller than the codomain (the set of all possible outputs).

We state the following: A function f: A → B is called a function not onto if there exists at least one element in set B that is not the image of any element in set A.

Key characteristics include the following:

• Range subset Codomain: The range is a proper subset of the codomain.

• Non-Exhaustive: Not every element in the target set is used.

• Comparison with Onto: Unlike an "onto" function (Surjective) where the range equals the codomain, this function always leaves something behind.

Representation Methods in Into Function

Understanding this function becomes much easier when you look at how it is represented in different formats.

• Arrow Diagram (Mapping Representation)
  In this method, elements of the domain are connected to elements of the codomain using arrows.
  If at least one element in the codomain has no arrow pointing to it, the function is an non-surjective function.

• Ordered Pairs Representation
  A function can also be written as a set of ordered pairs like:
  {(1, 4), (2, 5), (3, 6)}
  Here, if the codomain includes an extra value like 7 that does not appear in any pair, the function is into.

• Algebraic Representation
  Functions written as formulas like f(x) = x² can also be Into, depending on the codomain.

These multiple representations help you identify this function across diagrams, equations, and graphs.

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Into Function Definition in Sets and Mapping 

Let’s look at this through a set-based lens. Suppose we have:

• Set A (Domain): {1, 2, 3}

• Set B (Codomain): {4, 5, 6, 7}

If the mapping is f(1) = 4, f(2) = 5, and f(3) = 6, then the element '7' in Set B is left without a pre-image in Set A. Because '7' is unused, this mapping is a classic example of a function in mathematics.

Properties of Into Functions

1. Subset Relation: The most defining property is that the range f(A) is strictly contained within the codomain B.

2. Existence of Pre-images: At least one element y in B exists such that no x in A satisfies f(x) = y.

3. Versatility: It can be "one-to-one" (injective) or "many-to-one", as long as it fails to cover the entire codomain.

Read More - One to One Function - Definition and Examples

Into Function Graph

Visualising these concepts helps solidify the theory. This graph typically reveals its nature through the vertical and horizontal relationships between the curve and the y-axis.

When looking at a graph:

• Vertical Line Test: Confirms it is a function (each x has only one y).

• Range Observation: If the codomain is defined as all real numbers (\mathbb{R}), but the graph only exists in a specific portion of the y-axis (for example, only positive values), it is an into-function.

Example: f(x) = x², where the codomain is all real numbers.

In this graph, the curve (a parabola) only occupies the upper half of the coordinate plane. Negative values on the y-axis are never reached. Since there are values in the codomain (like -2 or -5) that have no corresponding x value, the function is "Into."

Vertical Line Test for Into Function

To confirm whether a graph represents a valid function before checking if it is into, we use the vertical line test.

Follow these steps:

1. Draw a vertical line anywhere on the graph.

2. Check how many points it intersects.

3. If the line intersects the graph at more than one point, it is not a function.

4. If it intersects at only one point everywhere, it is a valid function.

Once confirmed as a function, you can then check whether it is a non-surjective function by analysing its range and codomain.

Into Function vs. Onto Function

It is easier to understand these mappings by comparing them side-by-side.

Into Function Examples

To get comfortable with the concept, let’s look at a few examples across different mathematical scenarios.

1. The Constant Function

Consider a function f: ℝ → ℝ defined by f(x) = 5.

In this case, every single input x maps to the number 5. The codomain is the set of all real numbers, but the range is just the single set {5}. Since billions of other real numbers are left unmapped, this is a clear example.

2. Squaring Real Numbers

Let f: \mathbb{Z} \rightarrow \mathbb{Z} be defined by f(x) = x^2.

If we take the domain and codomain as integers:

• Inputs: {-2, -1, 0, 1, 2}

• Outputs: {4, 1, 0, 1, 4}

• The codomain contains negative integers like -1, -2, -3.
  Since no integer squared results in a negative number, those negative values are "left out", making it a non-surjective function,

3. Absolute Value Function

Let f: ℝ → ℝ be defined by f(x) = |x

The absolute value of any number is always non-negative. However, the codomain is defined as all real numbers (including negatives). Because the range [0, \infty) does not cover the entire codomain (-\infty, \infty), this qualifies under the non-surjective function.

Read More - Function Formulas – List of Key Function Formulas

How to Identify a Non-Surjective function in Maths ?

To check if a function is "into," follow these steps:

1. Identify the Codomain: This is usually given in the function notation (e.g., f: A → B, where B is the codomain).

2. Find the Range: Solve for the possible values of y that the function can produce.

3. Compare: If you find even one value in the codomain that cannot be produced by the function.

Practice Questions for Into Function

Let’s test your understanding with exam-style questions.

Question 1:
Let f: {1, 2, 3} → {4, 5, 6, 7} be defined as
f(1)=4, f(2)=5, f(3)=6
Is this a non-surjective function?

Solution:
The codomain contains 7, which has no pre-image.
Therefore, it is a non-surjective function.

Question 2:
Let f: ℝ → ℝ be defined as f(x) = x²
Is this a non-surjective function?

Solution:
The range is [0, ∞), while the codomain is all real numbers.
Negative values are not covered.
Therefore, it is a non-surjective function.

Non-Surjective Function Short Summary 

Here is a quick summary of the main properties of non-surjective function

• It does not cover the entire codomain

• The range is always a proper subset of the codomain

• It can be one-to-one or many-to-one

• Not all relations are functions, so check validity first

• Graphically, missing y-values indicate a non-surjective function.

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