How To Prove A Function Is One To One

In the realm of mathematics, particularly in calculus and analysis, the concept of a one-to-one function, also known as an injective function, holds significant importance. A function is considered one-to-one if each element of the range is associated with exactly one element of the domain. This property ensures that no two distinct elements in the domain map to the same element in the range. Proving that a function is one-to-one is a fundamental skill that underlies many mathematical proofs and applications.

Understanding One-to-One Functions

A function f is one-to-one (or injective) if for every x₁ and x₂ in its domain, if f(x₁) = f(x₂), then x₁ = x₂. In simpler terms, if different inputs always produce different outputs, the function is one-to-one.

Why is this important?

• Inverse Functions: A function has an inverse if and only if it is one-to-one. The inverse function essentially "undoes" what the original function does, and this is only possible if each output corresponds to a unique input.
• Mathematical Proofs: The one-to-one property is often used in mathematical proofs to establish uniqueness or to demonstrate that certain mappings preserve distinctness.
• Computer Science: In computer science, injective functions are used in hashing algorithms, cryptography, and data compression to ensure data integrity and efficient storage.

Methods to Prove a Function is One-to-One

There are several methods to prove that a function is one-to-one, each with its own advantages and disadvantages. The choice of method depends on the nature of the function and the tools available. Here, we will explore the most common techniques:

1. Direct Proof (Algebraic Method)
2. Proof by Contradiction
3. Using the Derivative (Calculus Method)
4. Graphical Method (Horizontal Line Test)

1. Direct Proof (Algebraic Method)

The direct proof is the most straightforward method and relies on the definition of a one-to-one function. It involves assuming that f(x₁) = f(x₂) and then algebraically manipulating the equation to show that x₁ must equal x₂.

Steps:

1. Assume f(x₁) = f(x₂) for arbitrary x₁ and x₂ in the domain of f.
2. Algebraically manipulate the equation f(x₁) = f(x₂).
3. Show that x₁ = x₂ through logical deduction.
4. Conclude that f is one-to-one.

Example 1:

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

Proof:

1. Assume f(x₁) = f(x₂) for some x₁, x₂ in the domain of f.
2. Then, 3x₁ + 5 = 3x₂ + 5.
3. Subtracting 5 from both sides gives 3x₁ = 3x₂.
4. Dividing both sides by 3 gives x₁ = x₂.
5. Therefore, f(x) is one-to-one.

Example 2:

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

Proof:

1. Assume f(x₁) = f(x₂) for some x₁, x₂ in the domain of f.
2. Then, x₁³ = x₂³.
3. Taking the cube root of both sides gives ∛(x₁³) = ∛(x₂³).
4. Thus, x₁ = x₂.
5. Therefore, f(x) is one-to-one.

Example 3:

Prove that f(x) = (x - 2) / (x + 3), for x ≠ -3, is a one-to-one function.

Proof:

1. Assume f(x₁) = f(x₂) for some x₁, x₂ in the domain of f.
2. Then, (x₁ - 2) / (x₁ + 3) = (x₂ - 2) / (x₂ + 3).
3. Cross-multiplying gives (x₁ - 2)(x₂ + 3) = (x₂ - 2)(x₁ + 3).
4. Expanding both sides gives x₁x₂ + 3x₁ - 2x₂ - 6 = x₁x₂ + 3x₂ - 2x₁ - 6.
5. Simplifying, we get 3x₁ - 2x₂ = 3x₂ - 2x₁.
6. Adding 2x₂ + 2x₁ to both sides gives 5x₁ = 5x₂.
7. Dividing both sides by 5 gives x₁ = x₂.
8. Therefore, f(x) is one-to-one.

Limitations:

• This method requires strong algebraic skills.
• It may not be suitable for all types of functions, especially those involving complex expressions or transcendental functions.

2. Proof by Contradiction

Proof by contradiction is an indirect method that assumes the opposite of what you want to prove and then shows that this assumption leads to a contradiction.

Steps:

1. Assume that f(x) is not one-to-one. This means there exist x₁ ≠ x₂ such that f(x₁) = f(x₂).
2. Show that this assumption leads to a contradiction (an impossible statement or a violation of given conditions).
3. Conclude that the initial assumption is false, and therefore f(x) must be one-to-one.

Example:

Prove that f(x) = √x for x ≥ 0 is one-to-one using proof by contradiction.

Proof:

1. Assume that f(x) is not one-to-one. This means there exist x₁ ≠ x₂ such that f(x₁) = f(x₂).
2. Then, √x₁ = √x₂.
3. Squaring both sides gives x₁ = x₂.
4. This contradicts our initial assumption that x₁ ≠ x₂.
5. Therefore, the assumption that f(x) is not one-to-one is false, and f(x) = √x is one-to-one.

Limitations:

• Finding the contradiction can sometimes be challenging.
• Requires a good understanding of logical reasoning.

3. Using the Derivative (Calculus Method)

If the function is differentiable, calculus provides a powerful tool to determine if it's one-to-one. The derivative of a function gives information about its rate of change.

Theorem:

• If f'(x) > 0 for all x in the domain, then f(x) is strictly increasing and hence one-to-one.
• If f'(x) < 0 for all x in the domain, then f(x) is strictly decreasing and hence one-to-one.

Steps:

1. Find the derivative f'(x) of the function f(x).
2. Show that f'(x) is either always positive or always negative over the entire domain.
3. Conclude that f(x) is either strictly increasing or strictly decreasing, and therefore one-to-one.

Example 1:

Prove that f(x) = x³ + 2x + 1 is one-to-one using the derivative.

Proof:

1. Find the derivative: f'(x) = 3x² + 2.
2. Since x² ≥ 0 for all x, it follows that 3x² ≥ 0. Thus, 3x² + 2 ≥ 2 > 0 for all x.
3. Since f'(x) > 0 for all x, f(x) is strictly increasing and therefore one-to-one.

Example 2:

Prove that f(x) = -x⁵ - x³ - x is one-to-one using the derivative.

Proof:

1. Find the derivative: f'(x) = -5x⁴ - 3x² - 1.
2. Since x⁴ ≥ 0 and x² ≥ 0 for all x, it follows that -5x⁴ ≤ 0 and -3x² ≤ 0. Thus, -5x⁴ - 3x² - 1 ≤ -1 < 0 for all x.
3. Since f'(x) < 0 for all x, f(x) is strictly decreasing and therefore one-to-one.

Limitations:

• This method requires the function to be differentiable.
• It only works if the derivative is strictly positive or strictly negative over the entire domain. If the derivative changes sign, the function may not be one-to-one.
• Some functions are difficult to differentiate.

4. Graphical Method (Horizontal Line Test)

The horizontal line test is a visual method for determining if a function is one-to-one.

The Horizontal Line Test:

A function f(x) is one-to-one if and only if no horizontal line intersects the graph of f(x) more than once.

Steps:

1. Graph the function f(x).
2. Draw horizontal lines across the graph.
3. Observe the number of times each horizontal line intersects the graph.
4. If no horizontal line intersects the graph more than once, then f(x) is one-to-one. Otherwise, f(x) is not one-to-one.

Example 1:

Consider the function f(x) = x². The graph of f(x) = x² is a parabola. Horizontal lines above the x-axis intersect the graph twice, meaning that f(x) = x² is not one-to-one over the entire real line. However, if we restrict the domain to x ≥ 0, the function f(x) = x² becomes one-to-one.

Example 2:

Consider the function f(x) = x³. The graph of f(x) = x³ is a cubic curve. Any horizontal line will intersect the graph exactly once, meaning that f(x) = x³ is one-to-one.

Limitations:

• This method is subjective and depends on the accuracy of the graph.
• It may not be suitable for complex functions where graphing is difficult.
• It's less rigorous than algebraic or calculus-based methods.

Examples and Case Studies

Let's explore additional examples to solidify our understanding.

Example 4: f(x) = eˣ

Prove that f(x) = eˣ is one-to-one.

Method 1: Direct Proof

1. Assume f(x₁) = f(x₂).
2. Then, e^(x₁) = e^(x₂).
3. Taking the natural logarithm of both sides gives ln(e^(x₁)) = ln(e^(x₂)).
4. Thus, x₁ = x₂.
5. Therefore, f(x) = eˣ is one-to-one.

Method 2: Using the Derivative

1. f'(x) = eˣ.
2. Since eˣ > 0 for all x, f'(x) > 0 for all x.
3. Therefore, f(x) = eˣ is strictly increasing and one-to-one.

Example 5: f(x) = sin(x) for 0 ≤ x ≤ π/2

Prove that f(x) = sin(x) is one-to-one on the interval [0, π/2].

Method: Using the Derivative

1. f'(x) = cos(x).
2. On the interval [0, π/2], cos(x) ≥ 0. More precisely, cos(x) > 0 for 0 ≤ x < π/2. At x = π/2, cos(x) = 0, but since the function is still increasing up to that point, we can consider it one-to-one.
3. Therefore, f(x) = sin(x) is strictly increasing on [0, π/2] and thus one-to-one.

Note: sin(x) is not one-to-one over its entire domain (-∞, ∞), as it oscillates between -1 and 1.

Example 6: f(x) = x² + 2x - 1 for x ≥ -1

Prove that f(x) = x² + 2x - 1 is one-to-one for x ≥ -1.

Method: Using the Derivative

1. f'(x) = 2x + 2.
2. For x ≥ -1, 2x ≥ -2, so 2x + 2 ≥ 0. More precisely, f'(x) > 0 for x > -1.
3. Therefore, f(x) = x² + 2x - 1 is strictly increasing for x ≥ -1 and thus one-to-one.

Example 7: Show that f(x) = x² is NOT one-to-one

Method: Direct Counterexample

Consider x₁ = 2 and x₂ = -2. Then, f(2) = 2² = 4 and f(-2) = (-2)² = 4. Thus, f(2) = f(-2) but 2 ≠ -2. Therefore, f(x) = x² is not one-to-one.

Case Study: Encryption Functions

In cryptography, one-to-one functions are crucial for creating secure encryption algorithms. A simple example is the Caesar cipher, where each letter in the alphabet is shifted by a fixed number of positions. If the shift is a valid one-to-one mapping, it can be easily decrypted by reversing the shift.

More complex encryption algorithms, like the Advanced Encryption Standard (AES), rely on multiple layers of substitution and permutation operations, all of which must be one-to-one to ensure that the encrypted data can be decrypted back to its original form.

Common Mistakes and Pitfalls

• Assuming instead of Proving: A common mistake is to assume that a function is one-to-one without providing a rigorous proof. Always use one of the methods described above to justify your claim.
• Incorrect Algebra: Algebraic errors can lead to incorrect conclusions. Double-check each step in the algebraic manipulation.
• Ignoring the Domain: The domain of a function can significantly affect whether it is one-to-one. A function may be one-to-one over a restricted domain but not over its entire domain.
• Misinterpreting the Derivative: Make sure to correctly interpret the derivative. f'(x) ≥ 0 (or f'(x) ≤ 0) does not necessarily imply the function is one-to-one. It must be strictly greater than (or less than) zero over the relevant interval (with possibly a finite number of points where the derivative is zero).
• Relying Solely on the Graph: While the horizontal line test is a helpful visual aid, it's not a substitute for a rigorous proof.

Conclusion

Proving that a function is one-to-one is a fundamental skill in mathematics with applications in various fields. Whether using the direct proof, proof by contradiction, the derivative, or the graphical method, understanding the underlying principles is crucial. By carefully applying these methods and avoiding common pitfalls, you can confidently determine whether a function is one-to-one and leverage this property in more advanced mathematical contexts. The ability to rigorously prove these fundamental properties underpins deeper understanding and proficiency in mathematical analysis and its applications. Remember that the choice of method depends on the function itself, and proficiency comes with practice and familiarity with a wide range of examples.