What Is an Injective Function Definition Properties and Solved Examples
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 Injective Function Explained with Definition and Proof
1. What is an injective function?
An injective function (also called a one-to-one function) is a function where different inputs produce different outputs. In other words, if f(a) = f(b), then it must follow that a = b.
- No two distinct elements in the domain map to the same element in the codomain.
- Each output is associated with at most one input.
- Injective functions may not use every element in the codomain.
2. How do you prove a function is injective?
To prove a function is injective, assume f(a) = f(b) and show that a = b. This method is called the direct proof of injectivity.
- Step 1: Assume f(a) = f(b).
- Step 2: Substitute the function definition.
- Step 3: Simplify the equation.
- Step 4: Conclude that a = b.
3. What is the horizontal line test for injective functions?
The horizontal line test states that a function is injective if every horizontal line intersects its graph at most once. If a horizontal line crosses the graph more than once, the function is not one-to-one.
- Draw horizontal lines across the graph.
- If any line intersects the graph at two or more points, it is not injective.
- If every line intersects once or not at all, it is injective.
4. What is the difference between injective and surjective functions?
An injective function maps distinct inputs to distinct outputs, while a surjective function covers every element in the codomain. The key difference lies in uniqueness versus completeness.
- Injective (one-to-one): No repeated outputs.
- Surjective (onto): Every codomain element has at least one preimage.
- A function that is both is called bijective.
5. Can you give an example of an injective function?
An example of an injective function is f(x) = 2x + 5 over the real numbers. This function is one-to-one because different values of x produce different outputs.
- If f(a) = f(b), then 2a + 5 = 2b + 5.
- Subtract 5: 2a = 2b.
- Divide by 2: a = b.
6. Is every linear function injective?
A linear function f(x) = mx + c is injective if and only if m ≠ 0. The slope determines whether the function is one-to-one.
- If m ≠ 0, the function is strictly increasing or decreasing, so it is injective.
- If m = 0, then f(x) = c (a constant function), which is not injective.
7. How do you check if a function is injective algebraically?
To check injectivity algebraically, set f(a) = f(b) and verify whether this implies a = b. This method works for polynomial, rational, and other algebraic functions.
- Write f(a) = f(b).
- Simplify using algebraic manipulation.
- If the result forces a = b, the function is injective.
8. Why are injective functions important?
Injective functions are important because they guarantee unique mappings and allow the existence of an inverse function on their range. Without injectivity, an inverse would not be well-defined.
- They ensure each output comes from exactly one input.
- An inverse function exists if the function is injective.
- They are used in algebra, calculus, and discrete mathematics.
9. What is the condition for a function to have an inverse?
A function has an inverse if and only if it is injective on its domain. Injectivity ensures each output corresponds to exactly one input.
- If the function is injective, its inverse exists on its range.
- If it is not injective, different inputs share the same output, so an inverse cannot be defined.
- Graphically, it must pass the horizontal line test.
10. Is the function f(x) = x² injective?
The function f(x) = x² is not injective over ℝ because different inputs can give the same output. For example:
- f(2) = 4
- f(−2) = 4
