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 fundamental concept in mathematics, particularly in areas like calculus, linear algebra, and abstract algebra. Understanding this concept is crucial for various applications, including determining the invertibility of functions and understanding the properties of transformations. This comprehensive guide will explore various methods and techniques to prove a function's injectivity, providing clear explanations and illustrative examples.

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, 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. Conversely, if x₁ ≠ x₂, then f(x₁) ≠ f(x₂).

This contrasts with a many-to-one function where multiple elements in the domain map to the same element in the codomain. A one-to-one function ensures a unique mapping from each element in the domain to an element in the codomain.

Methods to Prove a Function is One-to-One

Several techniques can be used to demonstrate that a function is one-to-one. The best approach often depends on the nature of the function itself.

1. Direct Proof using the Definition

The most straightforward approach involves directly applying the definition of a one-to-one function. This involves assuming f(x₁) = f(x₂) and then showing that this implies x₁ = x₂.

Example:

Let's consider the function f(x) = 2x + 1. To prove it's one-to-one:

1. Assume: f(x₁) = f(x₂)
2. Substitute: 2x₁ + 1 = 2x₂ + 1
3. Simplify: 2x₁ = 2x₂
4. Solve for x: x₁ = x₂

Since we've shown that f(x₁) = f(x₂) implies x₁ = x₂, the function f(x) = 2x + 1 is one-to-one.

2. Using the Contrapositive

A slightly different approach involves using the contrapositive of the definition. The contrapositive states: If x₁ ≠ x₂, then f(x₁) ≠ f(x₂). Proving the contrapositive is logically equivalent to proving the original statement.

Example:

Let's use the same function, f(x) = 2x + 1.

1. Assume: x₁ ≠ x₂
2. Multiply by 2: 2x₁ ≠ 2x₂
3. Add 1: 2x₁ + 1 ≠ 2x₂ + 1
4. Substitute: f(x₁) ≠ f(x₂)

Therefore, since x₁ ≠ x₂ implies f(x₁) ≠ f(x₂), the function is one-to-one.

3. Graphical Method (Horizontal Line Test)

For functions represented graphically, the horizontal line test provides a quick visual check. If any horizontal line intersects the graph of the function at more than one point, the function is not one-to-one. If every horizontal line intersects the graph at most once, the function is one-to-one. This method is intuitive but not a formal proof.

4. Using Derivatives (for differentiable functions)

For differentiable functions, the monotonicity (strictly increasing or strictly decreasing) of the function can be used to determine injectivity. If the derivative f'(x) is always positive (or always negative) on the function's domain, then the function is strictly increasing (or strictly decreasing) and therefore one-to-one.

Example:

Consider the function f(x) = x³. Its derivative is f'(x) = 3x², which is always non-negative. While not strictly positive everywhere (it's zero at x=0), it is always positive except at x = 0. However, since x³ is strictly increasing on its entire domain (all real numbers), it is still one-to-one. Care must be taken with this method as the derivative being always positive isn't strictly sufficient for all cases.

5. Injective Functions Composed with Other Functions

If you have two injective functions, their composition is also injective. This property can simplify proofs.

Theorem: If f: A → B and g: B → C are injective functions, then the composition g ◦ f: A → C is also injective.

Proof:

Let's assume that (g ◦ f)(x₁) = (g ◦ f)(x₂). This means g(f(x₁)) = g(f(x₂)). Since g is injective, we can conclude that f(x₁) = f(x₂). Furthermore, since f is injective, we can conclude that x₁ = x₂. Therefore, g ◦ f is injective.

Examples of Proving Injectivity for Different Function Types

Let's explore more complex examples to solidify your understanding:

Example 1: Polynomial Functions

Consider the function f(x) = x³ - 3x + 5. To prove its injectivity, we can use the derivative. f'(x) = 3x² - 3. This is not always positive. Let’s consider different intervals:

• x > 1: f'(x) > 0 (function is increasing)
• x < -1: f'(x) > 0 (function is increasing)
• -1 < x < 1: f'(x) < 0 (function is decreasing)

The derivative is not always positive, which means we cannot directly use the derivative test to determine injectivity. A more detailed analysis, perhaps involving the second derivative or other techniques, would be needed to determine injectivity in this case. Simple graphical analysis shows this function is not one-to-one.

Example 2: Exponential Functions

Consider f(x) = eˣ. We can prove its injectivity using the derivative. f'(x) = eˣ, which is always positive for all real x. Therefore, f(x) = eˣ is strictly increasing and hence one-to-one.

Example 3: Trigonometric Functions

Trigonometric functions are not typically one-to-one over their entire domains. However, by restricting their domains, we can create injective versions. For example, sin(x) is injective on the interval [-π/2, π/2]. Proving this would require showing that if sin(x₁) = sin(x₂) for x₁, x₂ ∈ [-π/2, π/2], then x₁ = x₂. This can be done using trigonometric identities and the properties of the sine function within this restricted domain.

Example 4: Functions with Multiple Variables

For functions of multiple variables, injectivity is more complex. Consider a function f(x, y) = (x + y, x - y). To show this is injective, assume f(x₁, y₁) = f(x₂, y₂). This implies:

x₁ + y₁ = x₂ + y₂ x₁ - y₁ = x₂ - y₂

Adding these equations gives 2x₁ = 2x₂, which implies x₁ = x₂. Subtracting the equations gives 2y₁ = 2y₂, implying y₁ = y₂. Therefore, f(x, y) is injective.

Conclusion: Mastering the Art of Proving Injectivity

Proving a function is one-to-one requires a thorough understanding of function properties and a systematic approach. While the direct method is the most fundamental, employing other techniques like the contrapositive, graphical analysis, or derivative analysis (for suitable functions) can significantly simplify the process. Remember that the best method often depends on the specific function being examined. Understanding these techniques allows you to tackle diverse functions, from simple linear equations to more complex multivariable and trigonometric functions, and confidently determine their injectivity. The ability to prove injectivity is a crucial skill in various mathematical fields and its mastery unlocks a deeper comprehension of function behavior and analysis.