How To Prove That A Function Is One To One

How to Prove That a Function is One-to-One (Injective)

Determining whether a function is one-to-one, also known as injective, is a fundamental concept in mathematics, particularly in areas like calculus, linear algebra, and abstract algebra. Understanding this concept is crucial for various applications, including cryptography, computer science, and optimization problems. This comprehensive guide will explore various methods to prove a function's injectivity, providing clear explanations, examples, and practical tips.

Understanding One-to-One Functions

A function, f: A → B, is considered one-to-one (or injective) if every element in the codomain B is mapped to by at most one element in the domain A. In simpler terms, each output value has a unique input value associated with it. No two distinct elements in the domain map to the same element in the codomain.

Key Characteristic: If f(x₁) = f(x₂), then x₁ = x₂. This is the core condition to demonstrate injectivity. The contrapositive statement is equally useful: If x₁ ≠ x₂, then f(x₁) ≠ f(x₂). Proving either statement proves the function is one-to-one.

Methods for Proving Injectivity

Several techniques can be employed to prove a function's injectivity, depending on the nature of the function:

1. Direct Proof using the Definition

This is the most straightforward approach. You directly apply the definition of injectivity. Assume f(x₁) = f(x₂), and then show through algebraic manipulation that this implies x₁ = x₂.

Example: Let's prove that f(x) = 3x + 5 is injective.

1. Assume: f(x₁) = f(x₂)
2. Substitute: 3x₁ + 5 = 3x₂ + 5
3. Simplify: Subtract 5 from both sides: 3x₁ = 3x₂
4. Solve for x: Divide both sides by 3: x₁ = x₂

Therefore, f(x) = 3x + 5 is injective.

Another Example (with a slight twist): Let's consider f(x) = x². This function is not injective over all real numbers because, for example, f(2) = f(-2) = 4. However, if we restrict the domain to non-negative real numbers (x ≥ 0), then it becomes injective. Let's prove this restricted function is injective.

1. Assume: f(x₁) = f(x₂) (where x₁ ≥ 0 and x₂ ≥ 0)
2. Substitute: x₁² = x₂²
3. Take the square root: √(x₁²) = √(x₂²)
4. Simplify: Since x₁ and x₂ are non-negative, |x₁| = |x₂| simplifies to x₁ = x₂.

2. Using the Contrapositive

This method utilizes the contrapositive of the definition of injectivity: If x₁ ≠ x₂, then f(x₁) ≠ f(x₂). It's often easier to prove the contrapositive than the original statement.

Example: Let's prove f(x) = eˣ is injective.

1. Assume: x₁ ≠ x₂
2. Consider the exponential function's properties: The exponential function is strictly increasing. This means that if x₁ < x₂, then eˣ¹ < eˣ², and if x₁ > x₂, then eˣ¹ > eˣ².
3. Conclusion: Since x₁ ≠ x₂, it follows that eˣ¹ ≠ eˣ².

Therefore, f(x) = eˣ is injective.

3. Graphical Method

For functions that can be easily graphed, the horizontal line test provides a visual way to determine injectivity. If any horizontal line intersects the graph of the function at more than one point, the function is not injective. If every horizontal line intersects the graph at most once, then the function is injective.

4. Calculus Approach (for differentiable functions)

If a function is differentiable, its derivative can be used to determine injectivity over an interval.

• Strictly Increasing/Decreasing Functions: If the derivative f'(x) > 0 for all x in an interval, then f(x) is strictly increasing and therefore injective on that interval. Similarly, if f'(x) < 0 for all x in an interval, f(x) is strictly decreasing and injective on that interval.

• Monotonicity: A function is monotonic if it is either entirely increasing or entirely decreasing. Monotonic functions are always injective (except for constant functions which are not injective).

Example: Consider f(x) = x³

1. Find the derivative: f'(x) = 3x²
2. Analyze the derivative: f'(x) ≥ 0 for all x (it's zero only at x=0). While not strictly greater than 0 for all x, it's non-negative, and the function is strictly increasing.
3. Conclusion: f(x) = x³ is injective over all real numbers.

5. Proving Injectivity for Composite Functions

If you have a composite function, g(f(x)), proving the injectivity of both g(x) and f(x) doesn't automatically mean g(f(x)) is injective. However, if both f(x) and g(x) are injective, then the composite function g(f(x)) is also injective.

Common Mistakes to Avoid

• Confusing injectivity with surjectivity: Injectivity deals with the uniqueness of the input, while surjectivity (or onto) deals with whether every element in the codomain is mapped to by at least one element in the domain. A function can be injective without being surjective, surjective without being injective, both, or neither.

• Incorrectly applying the horizontal line test: Remember to only use the horizontal line test for functions that can be readily graphed.

• Assuming injectivity based on limited observation: Don't assume a function is injective just because you've checked a few points. A rigorous proof is needed.

• Ignoring the domain and codomain: The injectivity of a function is often dependent on the specified domain and codomain.

Advanced Considerations

• Injectivity in multivariable calculus: The concept of injectivity extends to functions of multiple variables. The approach involves considering partial derivatives and Jacobian matrices.

• Injectivity and linear transformations: In linear algebra, injective linear transformations are characterized by having a trivial null space (only the zero vector maps to the zero vector).

Conclusion

Proving a function is one-to-one requires a careful and methodical approach. By understanding the various techniques outlined in this guide, including direct proof, the contrapositive, graphical methods, and the calculus approach, you can confidently determine the injectivity of a wide range of functions. Remember to always consider the function's domain and codomain, and avoid common pitfalls. Mastering this concept is essential for a strong foundation in mathematics and its applications. This comprehensive guide provides you with the tools to tackle various problems related to proving the injectivity of functions. Consistent practice and understanding the underlying principles will significantly enhance your ability to solve such problems effectively.