Functions are an important concept in mathematics that are used to describe relationships between sets. They allow us to model and analyze various phenomena in the real world, from the movement of planets to the behavior of financial markets. One of the most important properties of a function is whether it is one-to-one and onto, as these properties have significant implications for the behavior of the function.

In this mathematics article, we will explore what it means for a function to be one-to-one and onto, one-to-one and onto function examples, and why understanding these properties is crucial in many mathematical fields.

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One-to-One Functions

A one-to-one function is a type of function that maps each element in the range to only one element in its domain, which means that the outputs are unique and never repeated. 

A function is said to be both one-to-one and onto if it satisfies both properties. In other words, each element in the domain is paired with a unique element in the range, and every element in the range is mapped to by at least one element in the domain. 

Graphically, a one-to-one and onto function is a function where every point in the range is covered by exactly one point in the domain, and no horizontal line intersects the graph more than once. 

How to Check One-to-One and Onto Function?

One-to-One and Onto Function Graph

To provide a visual understanding of one-to-one and onto functions, here are the graphical representations of each:

One-to-One Function Graph:

The horizontal line test is used to determine whether a function is one-one when its graph is given. to test whether the function is one-one from its graph,

• To determine if a function is one-to-one, imagine a horizontal line (like a stick) passing through the graph. 
• If the line never intersects more than one point on the graph at any given moment, then the function is one-to-one. 
• However, if the line does intersect multiple points of the graph at any instance, then the function is not one-to-one.

For example, Consider the graph below.

Onto Function Graph:

To determine if a function is onto using its graph, a straightforward approach is to compare the range with the codomain. If the range is equal to the codomain, then the function is onto. For any function graph, it can be considered onto only if every horizontal line intersects the graph at one or more points. If there exists an element in the range that fails to intersect the function's graph when subjected to the horizontal line test, then the function is not onto. The following image exemplifies a graph of an onto function:

Difference Between One-to-One and Onto Function

Here's a tabular format highlighting the key differences between one-to-one and onto functions:

One-to-One and Onto Function Solved Examples

1.Determine if the function \(f(x) = 2x - 3\) is one-to-one or onto.

Solution:

To check if \(f(x)\) is one-to-one, assume that \(f(a) = f(b)\), where \(a\) and \(b\) are two different values in the domain of \(f(x)\). Then we have:

\(2a - 3 = 2b - 3\)

\(2a = 2b\)

\(a = b\)

Since \(a\) and \(b\) are equal, we can conclude that the function \(f(x)\) is one-to-one.

To check if \(f(x)\) is onto, we need to determine if every element in the range of \(f(x)\) has a corresponding element in the domain. The range of \(f(x)\) is all real numbers, since any real number can be obtained by plugging in a value for \(x\). 

Therefore, f(x) is onto.

2.Determine if the function \(g(x) = x^{2} - 4x\) is one-to-one or onto.

Solution:

To check if \(g(x)\) is one-to-one, we assume that \(g(a) = g(b)\), where \(a\) and \(b\) are two different values in the domain of \(g(x)\). Then we have:

\(a^{2} - 4a = b^{2} - 4b\)

\(a^{2} - b^{2} - 4a + 4b = 0\)

\((a - b)(a + b) - 4(a - b) = 0\)

\((a - b)(a + b - 4) = 0\)

Since \((a - b)(a + b - 4) = 0\), either \(a - b = 0\) or \(a + b - 4 = 0\). If \(a - b = 0\), then \(a = b\), and we have proven that \(g(x)\) is one-to-one. However, if \(a + b - 4 = 0\), then we cannot conclude that \(g(x)\) is one-to-one.

To check if \(g(x)\) is onto, we need to determine if every element in the range of \(g(x)\) has a corresponding element in the domain. The range of \(g(x)\) is all real numbers greater than or equal to \(-4\). However, there is no real number that can be plugged in for \(x\) to obtain a negative number in the range of \(g(x)\). 

Therefore, \(g(x)\) is not onto.

3.Determine if the function \(h(x) = 3x - 7\) is one-to-one or onto.

Solution:

To check if \(h(x)\) is one-to-one, assume that \(h(a) = h(b)\), where \(a\) and \(b\) are two different values in the domain of \(h(x)\). Then we have:

\(3a - 7 = 3b - 7\)

\(3a = 3b\)

\(a = b\)

Since \(a\) and \(b\) are equal, we can conclude that the function \(h(x)\) is one-to-one.

To check if \(h(x)\) is onto, we need to determine if every element in the range of \(h(x)\) has a corresponding element in the domain. The range of \(h(x)\) is all real numbers, since any real number can be obtained by plugging in a value for \(x\). 

Therefore, \(h(x)\) is onto.

Properties of a One-One Function

• Composition Rule:

If both functions f and g are one-one (injective), then their combination f ∘ g is also one-one.

1. Reverse Rule:
   If g ∘ f is one-one, then f must be one-one. But g may or may not be one-one.

2. Unique Mapping:
   A function f: X → Y is one-one if, for any two functions g and h (from a set P to X),
   if f ∘ g = f ∘ h, then it means g = h.
   This shows that one-one functions are special because they preserve uniqueness.

3. Image and Pre-image:
   If f: X → Y is one-one and P is a part (subset) of X,
   then we can get back P from its image: f⁻¹(f(P)) = P

4. Intersection Rule:
   If f is one-one and P and Q are two subsets of X, then: f(P ∩ Q) = f(P) ∩ f(Q)

5. Finite Set Rule:
   If X and Y have the same number of elements (finite sets),
   then a function f: X → Y is one-one if and only if it is also onto (surjective).

We hope that the above article is helpful for your understanding and exam preparations. Stay tuned to the Testbook App for more updates on related topics from Mathematics and various such subjects. Also, reach out to the test series available to examine your knowledge regarding several exams. For better practice, solve the below provided previous year papers and mock tests for each of the given entrance exam: