What is a One-to-One Function?

S3-SA5-0013

Grade Level:

Class 9

AI/ML, Data Science, Physics, Economics, Cryptography, Computer Science, Engineering

Definition

What is it?

A one-to-one function, also called an injective function, is a special type of function where every unique input value (from the domain) gives a unique output value (in the codomain). This means no two different inputs will ever produce the same output.

Simple Example

Quick Example

Imagine a classroom where each student has a unique roll number. If we define a function that maps each student to their roll number, it's a one-to-one function because no two students can have the same roll number. Each student gets exactly one unique roll number.

Worked Example

Step-by-Step

Let's check if the function f(x) = 2x + 3 is one-to-one. --- Step 1: Assume that for two different inputs, 'a' and 'b', their outputs are the same. So, f(a) = f(b). --- Step 2: Substitute 'a' and 'b' into the function: 2a + 3 = 2b + 3. --- Step 3: Subtract 3 from both sides: 2a = 2b. --- Step 4: Divide both sides by 2: a = b. --- Step 5: Since we started by assuming f(a) = f(b) and it led to a = b, this means the only way for the outputs to be the same is if the inputs were already the same. --- Answer: Therefore, f(x) = 2x + 3 is a one-to-one function.

Why It Matters

One-to-one functions are super important in fields like computer science for secure data encryption, ensuring every piece of information has a unique code. In data science, they help create unique identifiers for objects. Understanding them can open doors to careers in cybersecurity and software development.

Common Mistakes

MISTAKE: Thinking that if two inputs give different outputs, it's one-to-one. | CORRECTION: A function is one-to-one if *no two different inputs* ever give the *same output*. It's about preventing duplicates in the output.

MISTAKE: Confusing one-to-one with 'onto' (surjective) functions. | CORRECTION: One-to-one means each input maps to a unique output. Onto means every element in the codomain is an output for at least one input.

MISTAKE: Only checking a few input values to decide if a function is one-to-one. | CORRECTION: You need to prove it for all possible inputs, usually by using algebraic methods like assuming f(a) = f(b) and showing a = b.

Practice Questions

Try It Yourself

QUESTION: Is the function g(x) = x^2 one-to-one? | ANSWER: No, because g(2) = 4 and g(-2) = 4. Different inputs (2 and -2) give the same output (4).

QUESTION: Determine if the function h(x) = x - 5 is one-to-one. Show your steps. | ANSWER: Assume h(a) = h(b). Then a - 5 = b - 5. Adding 5 to both sides gives a = b. Since a = b, the function h(x) = x - 5 is one-to-one.

QUESTION: A function assigns each student in a class to their unique Aadhaar number. Is this a one-to-one function? Explain why. | ANSWER: Yes, this is a one-to-one function. Each student has only one Aadhaar number, and crucially, no two students can share the same Aadhaar number. So, unique students map to unique Aadhaar numbers.

MCQ

Quick Quiz

Which of the following functions is a one-to-one function?

f(x) = |x|

g(x) = x^2 + 1

h(x) = 3x

k(x) = 5

The Correct Answer Is:

For h(x) = 3x, if 3a = 3b, then a = b, meaning unique inputs always give unique outputs. For options A and B, different inputs like -2 and 2 give the same output. For option D, all inputs give the same output.

Real World Connection

In the Real World

Think about your mobile phone's fingerprint scanner or face unlock. This technology relies on one-to-one mapping – your unique fingerprint or face is mapped to your identity. No two people can unlock your phone with their unique biometrics, ensuring security. This is a real-world application of one-to-one functions.

Key Vocabulary

Key Terms

FUNCTION: A rule that assigns each input to exactly one output. | DOMAIN: The set of all possible input values for a function. | CODOMAIN: The set of all possible output values for a function. | UNIQUE: Being the only one of its kind; distinct from all others.

What's Next

What to Learn Next

Great job understanding one-to-one functions! Next, you should explore 'onto functions' (surjective functions) and then 'bijective functions' which are both one-to-one and onto. This will give you a complete picture of different types of functions.