Proof Of One To One Function

Proving One-to-One Functions: A Comprehensive Guide

Understanding one-to-one functions, also known as injective functions, is crucial in various areas of mathematics, particularly in calculus, linear algebra, and abstract algebra. A function is considered one-to-one if every element in the range corresponds to exactly one element in the domain. In simpler terms, no two distinct elements in the domain map to the same element in the range. This article provides a comprehensive guide to proving whether a function is one-to-one, encompassing various techniques and illustrative examples.

What is a One-to-One Function?

A function, denoted as f: A → B, is a one-to-one function (or injection) if for every x₁ and x₂ in the domain A, if f(x₁) = f(x₂), then x₁ = x₂. Alternatively, and equivalently, if x₁ ≠ x₂, then f(x₁) ≠ f(x₂). This means each output value in the range B corresponds to only one input value in the domain A.

Key takeaway: The core idea is to demonstrate that distinct inputs always produce distinct outputs.

Methods for Proving One-to-One Functions

Several methods can be employed to prove a function is one-to-one. The most common approaches include:

1. Direct Proof using the Definition

This is the most straightforward method. You directly apply the definition of a one-to-one function. Start by assuming f(x₁) = f(x₂), then manipulate the equation algebraically to show that x₁ = x₂.

Example:

Prove that f(x) = 3x + 5 is a one-to-one function.

Proof:

Let x₁ and x₂ be elements in the domain of f. Assume f(x₁) = f(x₂). Then:

3x₁ + 5 = 3x₂ + 5

Subtracting 5 from both sides:

3x₁ = 3x₂

Dividing both sides by 3:

x₁ = x₂

Therefore, if f(x₁) = f(x₂), then x₁ = x₂. This proves that f(x) = 3x + 5 is a one-to-one function.

2. Contrapositive Proof

This method uses the contrapositive of the definition. The contrapositive of "If P, then Q" is "If not Q, then not P." In the context of one-to-one functions, the contrapositive is: "If x₁ ≠ x₂, then f(x₁) ≠ f(x₂)."

Example:

Prove that f(x) = x³ is a one-to-one function.

Proof:

Let x₁ and x₂ be elements in the domain of f. Assume x₁ ≠ x₂. Then, cubing both sides, we get x₁³ ≠ x₂³. Therefore, f(x₁) ≠ f(x₂). This proves that f(x) = x³ is a one-to-one function using the contrapositive.

3. Graphical Method (Horizontal Line Test)

This method is visual and intuitive, but it's less rigorous and only applicable for functions with easily-graphable forms. If every horizontal line intersects the graph of the function at most once, the function is one-to-one. If a horizontal line intersects the graph more than once, the function is not one-to-one.

Example:

Consider the function f(x) = x². The graph is a parabola. A horizontal line above the x-axis intersects the parabola at two points. Therefore, f(x) = x² is not a one-to-one function. However, if we restrict the domain to x ≥ 0, then the horizontal line test works, and the function becomes one-to-one.

4. Using Calculus (Monotonic Functions)

For differentiable functions, if the function is strictly increasing or strictly decreasing over its entire domain, it's one-to-one. This is because a strictly monotonic function never takes on the same value twice. To check for monotonicity, analyze the derivative:

• Strictly Increasing: f'(x) > 0 for all x in the domain.
• Strictly Decreasing: f'(x) < 0 for all x in the domain.

Example:

Prove that f(x) = eˣ is a one-to-one function.

Proof:

The derivative of f(x) = eˣ is f'(x) = eˣ. Since eˣ > 0 for all x, f(x) is strictly increasing. Therefore, f(x) = eˣ is a one-to-one function.

Examples of One-to-One and Not One-to-One Functions

Let's explore more examples to solidify our understanding:

One-to-One Functions:

• f(x) = 2x - 7: A linear function with a non-zero slope is always one-to-one.
• f(x) = x³ + 1: A cubic function (without repeated roots) is one-to-one.
• f(x) = ln(x) (for x > 0): The natural logarithm function is one-to-one for positive x values.
• f(x) = tan⁻¹(x): The inverse tangent function is one-to-one.

Not One-to-One Functions:

• f(x) = x²: As shown earlier, a parabola fails the horizontal line test.
• f(x) = sin(x): The sine function is periodic, meaning it repeats its values, making it not one-to-one.
• f(x) = |x|: The absolute value function maps both positive and negative values to their positive counterparts.
• f(x) = x⁴: Similar to x², this function fails the horizontal line test.

Importance of One-to-One Functions

One-to-one functions are fundamental in several mathematical contexts:

• Inverse Functions: Only one-to-one functions have inverse functions. The inverse function essentially "reverses" the mapping of the original function.
• Bijections: A bijection is a function that is both one-to-one and onto (surjective). Bijections are crucial in establishing correspondences between sets and are essential in many areas of mathematics, including set theory and abstract algebra.
• Cryptography: One-to-one functions play a vital role in cryptography, where they ensure that different inputs produce unique outputs, essential for secure encryption and decryption.
• Calculus: Understanding one-to-one functions is vital for finding the inverse of functions and applying the inverse function theorem.
• Linear Algebra: One-to-one linear transformations are crucial for studying the properties of vector spaces and linear systems.

Advanced Topics and Further Exploration

This article provided a foundational understanding of proving one-to-one functions. For deeper exploration, consider these advanced topics:

• One-to-one correspondence (bijections): Explore functions that are both one-to-one and onto.
• Cardinality: Learn how one-to-one correspondences are used to compare the sizes of infinite sets.
• Injective functions in category theory: Delve into the abstract algebraic setting where injectivity is a fundamental concept.
• Applications in computer science: Explore how one-to-one functions are used in data structures, algorithms, and cryptography.

By mastering the techniques outlined in this article, you'll be equipped to confidently prove the one-to-one nature of various functions and appreciate their significance in diverse mathematical and computational contexts. Remember to practice regularly with different function types to solidify your understanding and improve your problem-solving skills. The ability to determine whether a function is one-to-one is a valuable skill for any student of mathematics or related fields.