How to Identify If a Function Is Injective or Surjective
The concept of injective function is essential in mathematics and helps in solving real-world and exam-level problems efficiently. Understanding injective functions (also called one-to-one functions) is key for topics like functions, domain and range, inverse functions, and bijective functions. Vedantu provides easy, student-friendly explanations with solved examples and clear visuals for injective functions.
Understanding Injective Function
An injective function is a type of function where each element in the domain maps to a unique element in the range. In other words, no two different inputs produce the same output. If \( f : X \rightarrow Y \) is injective, then for every pair of distinct elements \( x_1, x_2 \in X \), we have \( f(x_1) \neq f(x_2) \). Injective functions are widely used in relations and functions, bijective functions, domain and range, and mapping problems in set theory.
Definition and Meaning of Injective Function
Injective function is also called a one-to-one function or injection. Its key properties include:
- Every element of the domain has a unique partner in the range.
- No two different domain values have the same output.
- The function can be written as \( f : X \rightarrow Y \), where if \( f(x_1) = f(x_2) \), then \( x_1 = x_2 \).
- All injective functions are not necessarily surjective (onto), but every bijective function is injective.
Formula Used in Injective Function
The standard formula for checking if a function \( f : X \rightarrow Y \) is injective:
If \( f(x_1) = f(x_2) \Rightarrow x_1 = x_2 \) for all \( x_1, x_2 \) in the domain X, then f is injective.
Alternatively, show that if \( x_1 \neq x_2 \), then \( f(x_1) \neq f(x_2) \).
Worked Example – Solving a Problem
Let’s check if \( f(x) = 2x + 3 \) is an injective function from real numbers to real numbers.
1. Start by assuming \( f(x_1) = f(x_2) \).2. Substitute the formula: \( 2x_1 + 3 = 2x_2 + 3 \).
3. Subtract 3 from both sides: \( 2x_1 = 2x_2 \).
4. Divide by 2: \( x_1 = x_2 \).
Since this result shows \( x_1 = x_2 \), the function is injective (one-to-one).
More Examples of Injective Functions
Example 1: The function that assigns to each student in a class their unique roll number is injective since no two students share the same roll number.
Example 2: For \( f(x) = x + 5 \), check with numbers 1, 2, 3:
- \( f(1) = 6 \), \( f(2) = 7 \), \( f(3) = 8 \).
Distinct input values map to distinct outputs—so it is injective.
How to Test Injectivity – Step-by-Step
1. Start by setting \( f(x_1) = f(x_2) \).2. Solve for \( x_1 \) and \( x_2 \).
3. If you always find \( x_1 = x_2 \), then the function is injective.
4. If you can find \( x_1 \neq x_2 \) with \( f(x_1) = f(x_2) \), it’s not injective.
A graph passes the horizontal line test (where any horizontal line cuts the graph at most once) if and only if the function is injective.
Graphical View of Injective Function
Visualizing injective functions often helps clear confusion. If you draw the graph of \( y = f(x) \) and no horizontal line cuts the graph at more than one point, the function is injective.
For example, the straight line \( y = 2x + 1 \) is injective (except vertical lines), while \( y = x^2 \) is not injective (since different x can produce the same y for a parabola).
Comparison: Injective vs Surjective vs Bijective Functions
Let’s compare injective, surjective, and bijective functions so you don’t get confused:
| Type | Definition | Example |
|---|---|---|
| Injective (One-to-One) | Each element of domain maps to unique range value | \( y = x + 2 \) |
| Surjective (Onto) | Every element of range has at least one pre-image | \( y = x^3 \) |
| Bijective | Both injective and surjective | \( y = x \) |
Practice Problems
- Determine if \( f(x) = x^3 \) is injective on real numbers.
- Is the function \( f(x) = x^2 \) injective on integers?
- Give an example of a function that is injective but not surjective.
- Show that the function \( f(x) = 5x - 1 \) is injective.
Common Mistakes to Avoid
- Confusing injective function with surjective or bijective functions.
- Not testing with proper formula—always check \( f(x_1) = f(x_2) \Rightarrow x_1 = x_2 \).
- Assuming every function is injective without checking.
Real-World Applications
Injective functions help in computer science for database keys, ticketing systems where each ticket is unique, and in mathematics when working with inverse functions and cryptography. At Vedantu, understanding injective mappings is crucial for tackling advanced math and competitive exams.
Quick FAQ: Injective Function
Q1: What is an injective function?
A: A function where every input has a unique output. No two different domain elements map to the same value.
Q2: How do you check if a function is injective?
A: If \( f(x_1) = f(x_2) \) leads you to \( x_1 = x_2 \), the function is injective.
Q3: What is the difference between injective and surjective function?
A: Injective is one-to-one, surjective is onto. Bijective is both.
We explored the idea of injective function, its formula, examples, testing, and why this concept is vital for mathematics and beyond. Practice more on injective functions with Vedantu’s resources and ace your exams!
Further Study Links
FAQs on What Is an Injective Function? Definition and Key Concepts
1. What is meant by an injective function?
An injective function, also known as a one-to-one function, is a function in which each element of the domain maps to a unique element in the codomain. In other words, no two different inputs have the same output. Symbolically, a function f: A → B is injective if, whenever f(x₁) = f(x₂), then x₁ = x₂.
2. What is a surjective and injective function?
Injective functions (one-to-one) ensure unique outputs for unique inputs, while surjective functions (onto) map every element of the codomain to at least one element in the domain. A function is both injective and surjective if each input has a unique output and every possible output is covered – such functions are called bijective.
3. How to know when a function is injective?
To check if a function is injective:
- For every pair of distinct inputs x₁ and x₂, check if f(x₁) ≠ f(x₂).
- If no two different inputs give the same output, the function is injective.
Alternatively, use the definition: if f(x₁) = f(x₂) implies x₁ = x₂, then the function is injective.
4. What is a surjective function?
A surjective function, also called an onto function, is a function where every element in the codomain has at least one pre-image in the domain. This means that the function’s output covers the entire codomain.
5. What is the difference between injective and surjective functions?
Injective functions assign every input a unique output (no duplicates), while surjective functions ensure that every possible output value is achieved by at least one input. An injective function may miss values in the codomain, but a surjective function cannot.
6. What is an example of an injective function?
Example: The function f(x) = 2x + 3, where x is a real number, is injective because each input x gives a unique output, and no two different values of x will produce the same result.
7. What is a bijective function?
A bijective function is both injective (one-to-one) and surjective (onto). This means every element in the codomain corresponds to exactly one element in the domain. Bijective functions have inverses that are also functions.
8. Is every injective function also surjective?
No, an injective function is not always surjective. A function can be one-to-one without covering the entire codomain. To be both, a function must be bijective.
9. What is the formula to check if a function is injective?
A function f is injective if whenever f(x₁) = f(x₂), then x₁ = x₂. Or, if x₁ ≠ x₂ ⇒ f(x₁) ≠ f(x₂). Check this condition for the given function.
10. Can a function be both injective and surjective?
Yes, a function that is both injective and surjective is called a bijective function. Such a function is both one-to-one and onto.
11. What is the graph of an injective function like?
The graph of an injective function never passes through the same output (y-value) more than once. For real functions, if you draw a horizontal line, it will intersect the graph at most once (Horizontal Line Test).
12. What is the other name for injective function, and what is it called in Hindi?
An injective function is also known as a one-to-one function. In Hindi, it is called एक-एक फलन (ek-ek phalan).
