How To Show F Is One To One

How to Show a Function is One-to-One (Injective)

Showing that a function is one-to-one, also known as injective, is a crucial concept in mathematics, particularly in areas like linear algebra, calculus, and abstract algebra. Understanding this concept is vital for comprehending more advanced topics such as inverse functions, bijections, and isomorphisms. This comprehensive guide will walk you through various methods and strategies to prove a function's injectivity, illustrating each approach with clear examples.

Understanding One-to-One Functions

A function f: A → B is said to be 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, no two distinct elements in the domain map to the same element in the codomain. This means that if f(x₁) = f(x₂), then it must be the case that x₁ = x₂. This is the core principle we'll use to construct our proofs.

Methods for Proving Injectivity

There are several established approaches to demonstrate that a function is one-to-one. Let's explore the most common and effective methods:

1. The Direct Method (Using the Definition)

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

Example 1: Prove that f(x) = 3x + 5 is one-to-one.

Proof:

Let x₁, x₂ ∈ ℝ (the real numbers). 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, f(x) = 3x + 5 is one-to-one.

Example 2: Prove that f(x) = x³ is one-to-one for x ∈ ℝ.

Proof:

Let x₁, x₂ ∈ ℝ. Assume f(x₁) = f(x₂). Then:

x₁³ = x₂³

Taking the cube root of both sides:

x₁ = x₂

Therefore, f(x) = x³ is one-to-one. Note that the cube root is a well-defined operation for real numbers.

2. The Contrapositive Method

This method leverages the contrapositive of the implication. The contrapositive of "If P, then Q" is "If not Q, then not P." In the context of injectivity, this translates to: "If x₁ ≠ x₂, then f(x₁) ≠ f(x₂)."

Example 3: Prove that f(x) = eˣ is one-to-one for x ∈ ℝ.

Proof:

Let x₁, x₂ ∈ ℝ. Assume x₁ ≠ x₂. We need to show that f(x₁) ≠ f(x₂).

Since the exponential function is strictly increasing, if x₁ ≠ x₂, then eˣ¹ ≠ eˣ². Therefore, f(x₁) ≠ f(x₂).

Thus, f(x) = eˣ is one-to-one.

3. Using the Derivative (for Differentiable Functions)

If a function is differentiable, its derivative can provide information about its injectivity. A strictly increasing or strictly decreasing function is always one-to-one. A strictly increasing function has a positive derivative, while a strictly decreasing function has a negative derivative.

Example 4: Prove that f(x) = x² is not one-to-one for x ∈ ℝ.

Proof:

The derivative of f(x) = x² is f'(x) = 2x. This is positive for x > 0 and negative for x < 0. The function is neither strictly increasing nor strictly decreasing over its entire domain. For instance, f(2) = f(-2) = 4. Therefore, f(x) = x² is not one-to-one for x ∈ ℝ. However, if we restrict the domain to x ≥ 0, then it is one-to-one.

Example 5: Prove that f(x) = x³ + 2x + 1 is one-to-one for x ∈ ℝ.

Proof:

The derivative is f'(x) = 3x² + 2. Since x² ≥ 0 for all x ∈ ℝ, f'(x) ≥ 2 > 0 for all x. The derivative is always positive, implying that the function is strictly increasing and therefore one-to-one.

4. Using the Monotonicity of the Function

A function is monotonic if it is either always increasing or always decreasing. Monotonic functions are always one-to-one.

Example 6: Consider the function f(x) = ln(x) with domain (0, ∞). The natural logarithm is a strictly increasing function on its domain, hence it is one-to-one.

Important Considerations and Common Pitfalls

• Domain and Codomain: The domain and codomain significantly impact whether a function is one-to-one. A function might be one-to-one on a restricted domain but not on its entire natural domain. Always clearly specify the domain.

• Visual Inspection: For simple functions, graphing them can be helpful to visually inspect whether it's one-to-one (passes the horizontal line test: no horizontal line intersects the graph more than once). However, this is not a rigorous proof.

• Incorrect Assumptions: Don't assume a function is one-to-one without a proper proof. Many functions that appear one-to-one at first glance are not.

• Restricting the Domain: If a function is not one-to-one over its entire domain, you might be able to restrict the domain to a subset where it becomes one-to-one. This is crucial when dealing with inverse functions.

Advanced Techniques and Applications

For more complex functions or those defined on more abstract sets, more sophisticated techniques may be necessary. These may involve:

• Set theory arguments: Utilizing properties of sets and mappings to prove injectivity directly from definitions.
• Linear algebra techniques: For linear transformations, injectivity is equivalent to having a trivial kernel (null space).
• Group theory: In group theory, injectivity of homomorphisms plays a crucial role in understanding group structures.

Conclusion

Proving a function is one-to-one is a fundamental skill in mathematics with far-reaching implications. By mastering the methods outlined in this guide—the direct method, the contrapositive method, using derivatives, and considering monotonicity—you will be well-equipped to tackle a wide range of problems involving injectivity and its related concepts. Remember to always clearly define the domain and codomain and choose the most appropriate method based on the function's properties. With practice, these techniques will become second nature, allowing you to confidently navigate the world of injective functions.