One to One Function

Introduction to Functions

A function is a special rule in mathematics where every input (also called a domain value) is paired with exactly one output (also called a range value). You can think of a function as a machine: when you feed it a value (input), it gives you one result (output).

Example: f(x) = 2x
This function doubles whatever number you give it.
If x = 1, f(x) = 2
If x = 3, f(x) = 6
Each input gives only one output - this is a valid function.

 

Table of Content

• Understanding One to One Function
• Graphical Interpretation - Horizontal Line Test
• Algebraic Test for One to One
• Inverse of One to One Functions
• Real-Life Applications of One to One Functions
• Worked Examples
• Practice Questions
• Conclusion
• FAQs on One to One Function

 

Understanding One to One Function

A function is called one to one (or injective) when different inputs always produce different outputs. In other words, if you pick two different numbers as inputs, you will always get two different results.

This is very useful when we want each output to be uniquely tied to one input - no duplication.

Formal Definition:
A function f is one to one if:
f(x₁) = f(x₂) ⇒ x₁ = x₂

This means: If the outputs are the same, then the inputs must also be the same. There cannot be two different x-values that map to the same y-value.

 

Graphical Interpretation - Horizontal Line Test

To visually check if a function is one to one, we use the Horizontal Line Test:
Draw horizontal lines across the graph of the function.
If every horizontal line intersects the graph at most once, the function is one to one.
If a horizontal line touches the graph more than once, it is not a one to one function.

Note: The Vertical Line Test is used to check if a relation is a function (each input gives one output). The Horizontal Line Test checks if a function is one to one.

Examples of One to One and Not One to One Functions:

1. f(x) = x + 3
    This function increases steadily. Each input gives a unique output.
    Passes the horizontal line test ⇒ one to one

2. f(x) = x²
    f(2) = 4 and f(-2) = 4. Two different inputs give the same output.
    Fails the horizontal line test ⇒ Not one to one

3. f(x) = 2x + 1
    This is a linear function with a positive slope.
    It is strictly increasing and never repeats outputs.
    ⇒ One to one

 

Algebraic Test for One to One

Another way to verify if a function is one to one is through algebra.
Steps:
1. Assume f(a) = f(b)
2. Use algebra to solve and show that this leads to a = b
3. If it does for all a and b, then the function is one-to-one

Example:
f(x) = 5x - 7
Assume f(a) = f(b)
⇒ 5a - 7 = 5b - 7
⇒ a = b
Since we proved a = b, the function is one-to-one.

 

Inverse of One to One Functions

Only one to one functions have inverses. An inverse function "reverses" the process of the original function.

If f is one-to-one, then an inverse function f⁻¹ exists such that:
f(f⁻¹(x)) = x
f⁻¹(f(x)) = x

This means if you apply f and then f⁻¹, you return to your original value.

Example:
f(x) = 3x + 2
This is a one-to-one function.
To find the inverse:
Let y = 3x + 2
Solve for x: y - 2 = 3x ⇒ x = (y - 2)/3
So, f⁻¹(x) = (x - 2)/3

 

Real-Life Applications of One to One Functions

• Cryptography: In encryption, a message must have a unique encoded form to ensure security. One-to-one functions guarantee that each input has a unique output.

• Database Systems: Mapping unique user IDs to usernames or data requires a one-to-one relation.

• Linear Algebra & Geometry: In transformations, injective mappings preserve uniqueness of points.

• Computer Science: Hashing algorithms benefit from one-to-one behavior for storing and retrieving data.

 

Worked Examples

1. Prove f(x) = 5x - 7 is one-to-one:
Assume f(a) = f(b)
⇒ 5a - 7 = 5b - 7
⇒ a = b
So, f is one-to-one.

2. Is f(x) = x² one-to-one?
Try f(2) = 4 and f(-2) = 4
Different inputs give the same output ⇒ Not one to one.

 

Practice Questions

1. Is f(x) = x³ one-to-one? Justify.
(Hint: x³ is strictly increasing – each input has a unique output.)

2. Check if f(x) = sin(x) is one-to-one in [0, 2π]
(Hint: sin(0) = 0 and sin(π) = 0 ⇒ Not one-to-one)

3. Prove or disprove: f(x) = 1/x is one-to-one for x ≠ 0
(Hint: f(a) = f(b) ⇒ 1/a = 1/b ⇒ a = b)

4. Find the inverse of f(x) = (x - 4)/3
Let y = (x - 4)/3 ⇒ x = 3y + 4 ⇒ f⁻¹(x) = 3x + 4

 

Conclusion

A one to one function ensures that each input has a unique output and vice versa. Use the horizontal line test or algebraic method to confirm. Only one to one functions have inverse functions.
These functions are crucial in real-world contexts like coding, databases, encryption, and mathematics.

 

Related Links:

Algebraic Identities - Learn and apply key Algebraic Identities to simplify and solve expressions easily.

Algebraic Expression - Understand how to form, simplify, and evaluate Algebraic Expressions with confidence.

 

Frequently Asked Questions on One to One Function

1. What is a one to one function?

Ans: A one-one function (also called an injective function) is a function where each input maps to a unique output.
In other words, no two different inputs produce the same output.
If f(a) = f(b) implies a = b, then the function is one-one.

 

2. How do you know if a function is 1 to 1?

Ans: To check if a function is one to one:

• Use the horizontal line test on the graph: If no horizontal line cuts the graph more than once, it’s one-one.

• Or use algebra: If f(a) = f(b) always leads to a = b, then the function is one-one.
  
  

 

3. What are examples of one-one?

Ans: Examples of one-one (injective) functions:

• f(x) = x + 3 → Every x gives a unique output

• f(x) = 2x → Linear functions with non-zero slope are one-one

• f(x) = x³ → Cubic function is one-one

• f(x) = eˣ → Exponential function is one-one
  
  

Non-examples:

• f(x) = x² → Not one-one (since f(2) = f(-2))
  
  

 

4. How to determine if a function is onto or one to one?

Ans:

• A function is one-to-one (injective) if no two inputs have the same output: f(a) = f(b) ⇒ a = b.

• A function is onto (surjective) if every element in the codomain has a pre-image in the domain.
  To determine:

• Injective: Check algebraically or use the horizontal line test.

• Surjective: Solve f(x) = y for all y in the codomain. If you can find at least one x for every y, it's onto.
  
  

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