How to Identify and Prove a One To One Function with Steps and Properties
The concept of one to one function plays a key role in mathematics and is widely applicable to both real-life situations and exam scenarios. Understanding this topic helps in advanced algebra, calculus, and computer science logic. It also builds a strong foundation for competitive exams like JEE and Olympiads.
What Is One to One Function?
A one to one function (also called an injective function) is a function in which every output value is paired with only one unique input value. This means that no two different elements in the domain map to the same element in the range. You’ll find this concept applied in areas such as inverse functions, types of functions, and data encryption.
Key Formula for One to One Function
Here’s the standard formula to check for a one to one function:
If \( f(x_1) = f(x_2) \Rightarrow x_1 = x_2 \), then f is a one to one function.
Cross-Disciplinary Usage
One to one function is not only useful in Maths but also plays an important role in Physics, Computer Science, and daily logical reasoning. For example, in computer science, one to one mapping helps in creating unique keys in databases. Students preparing for JEE or NEET will see the relevance of this topic in questions about relations and functions and onto functions.
Step-by-Step Illustration
- Suppose \( f(x) = 2x + 3 \). Is this function one to one?
2. So, \( 2a + 3 = 2b + 3 \)
3. Subtract 3 from both sides: \( 2a = 2b \)
4. Divide by 2: \( a = b \)
Conclusion: Since a = b, the function is one to one.
Speed Trick or Vedic Shortcut
A quick way to test if a function is one to one is the horizontal line test. On the graph of a function, if any horizontal line meets the curve at most once, then the function is one to one. This is a popular shortcut in exams.
Example Trick: The function \( f(x) = x^2 \) fails the horizontal line test because the line y = 4 cuts the parabola at both x = 2 and x = -2. So, it’s not a one to one function!
Shortcuts like this save time in competitive exams. Vedantu’s live sessions provide such tips and tests to help students become accurate and fast.
Try These Yourself
- Check if \( f(x) = 3x - 7 \) is a one to one function.
- Is \( f(x) = x^2 \) a one to one function on all real numbers? What about on \( x \geq 0 \)?
- Draw the graph of \( y = x^3 \) and perform the horizontal line test.
- Given the set { (2,5), (3,6), (7,8) }, is this relation a one to one function?
Frequent Errors and Misunderstandings
- Confusing one to one function with one to many or many to one mappings.
- Assuming a function that passes the vertical line test is also one to one.
- Forgetting to check the domain when testing if a function is one to one (e.g., \( x^2 \) is one to one on x ≥ 0 but not on all real numbers).
One to One vs Many to One: Quick Comparison
| Function Type | Definition | Example |
|---|---|---|
| One to One | Every output has exactly one unique input | \( f(x) = x + 5 \) |
| Many to One | Multiple inputs have the same output | \( f(x) = x^2 \) (since 2 and -2 both map to 4) |
Relation to Other Concepts
The idea of one to one function connects closely with topics such as inverse functions (only one to one functions have inverses) and bijective functions (which are both one to one and onto). Mastering this concept helps with understanding advanced function properties and composition of functions in higher maths.
Classroom Tip
A quick way to remember one to one functions: Each y comes from only one x. Draw arrows in a set diagram—if two arrows ever land on the same output, it’s not one to one! Vedantu’s teachers often use coloured card activities in live classes to make this simple and visual.
Wrapping It All Up
We explored one to one function—from definition, key tests like the horizontal line test, quick tricks, worked examples, and links to other key Math concepts. Continue practicing with Vedantu’s problem sets and interactive sessions to become confident in solving questions on one to one functions and their inverses.
Explore related concepts: Onto Function | Inverse Function | Types of Relations | Types of Functions | Bijective Function
FAQs on One To One Function Explained with Definition and Examples
1. What is a one to one function?
A one to one function (also called an injective function) is a function in which each element of the range corresponds to exactly one element of the domain. In other words, if f(a) = f(b), then a = b.
- No two different inputs have the same output.
- Every output value is unique.
- It is also known as an injective mapping in mathematics.
2. How do you know if a function is one to one?
A function is one to one if different inputs always produce different outputs, meaning f(a) = f(b) implies a = b.
- Algebraically: Assume f(a) = f(b) and prove a = b.
- Graphically: Use the Horizontal Line Test.
- If any horizontal line cuts the graph more than once, it is not one to one.
3. What is the horizontal line test for a one to one function?
The horizontal line test states that a function is one to one if every horizontal line intersects its graph at most once.
- Draw horizontal lines across the graph.
- If any line touches the graph more than once, the function is not one to one.
- If all lines intersect once or not at all, the function is injective.
4. What is an example of a one to one function?
An example of a one to one function is f(x) = 2x + 3 because each input gives a unique output.
- Let f(a) = f(b).
- 2a + 3 = 2b + 3
- 2a = 2b
- a = b
5. What is an example of a function that is not one to one?
An example of a function that is not one to one is f(x) = x² because different inputs can give the same output.
- f(2) = 4
- f(−2) = 4
6. What is the difference between one to one and onto function?
A one to one function has unique outputs for each input, while an onto function (surjective) covers every element of the codomain.
- One to one: No repeated outputs.
- Onto: Every element in the codomain has at least one pre-image.
- A function that is both is called bijective.
7. Can a quadratic function be one to one?
A quadratic function is not one to one over all real numbers, but it can be one to one if its domain is restricted.
- For example, f(x) = x² is not injective on ℝ.
- If the domain is restricted to x ≥ 0, then it becomes one to one.
- Domain restriction ensures unique outputs.
8. Why is a one to one function important?
A one to one function is important because only injective functions have a true inverse function without restricting the domain.
- If a function is one to one, its inverse exists.
- Inverse functions undo the original function.
- They are widely used in algebra, calculus, and real-life modeling.
9. How do you prove a function is one to one algebraically?
To prove a function is one to one algebraically, assume f(a) = f(b) and show that a = b.
- Step 1: Start with f(a) = f(b).
- Step 2: Substitute the function formula.
- Step 3: Simplify the equation.
- Step 4: If you get a = b, the function is injective.
10. What is the condition for a function to be one to one?
The condition for a function to be one to one is that f(a) = f(b) implies a = b for all elements in the domain.
- Each output must correspond to exactly one input.
- No two distinct inputs share the same output.
- This condition defines an injective mapping in mathematics.
