How To Prove A Function Is Injective

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

This article will guide you through different methods of proving a function is injective, also known as one-to-one. Understanding injectivity is crucial in various areas of mathematics, including calculus, linear algebra, and abstract algebra. A function is injective if every element in the codomain is mapped to by at most one element in the domain. In simpler terms, no two distinct elements in the domain map to the same element in the codomain. Let's explore several techniques to demonstrate this property.

Understanding the Definition of Injectivity

Before delving into proof methods, let's solidify the definition. A function f: A → B is injective if and only if for all x, y ∈ A, if f(x) = f(y), then x = y. This means if the outputs are equal, then the inputs must also be equal. The contrapositive statement is equally useful: if x ≠ y, then f(x) ≠ f(y). This states that if the inputs are different, then the outputs must also be different. We will use both versions of the definition in the examples below.

Method 1: Direct Proof Using the Definition

This is the most straightforward approach. We start by assuming f(x) = f(y) for arbitrary x and y in the domain, and then we manipulate the equation to show that x = y.

Example: Prove that f(x) = 3x + 5 is injective.

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

Therefore, since f(x) = f(y) implies x = y, the function f(x) = 3x + 5 is injective.

Method 2: Proof by Contraposition

This method uses the contrapositive of the definition. We assume x ≠ y and show that f(x) ≠ f(y).

Example: Prove that g(x) = x² is not injective (for the domain of all real numbers).

1. Assume: x ≠ y (Let's choose x = 2 and y = -2 as a counterexample)
2. Evaluate: g(2) = 4 and g(-2) = 4
3. Conclusion: Since x ≠ y but g(x) = g(y), the function is not injective. Note that for the domain of non-negative real numbers, this function would be injective. The domain is crucial when determining injectivity.

Method 3: Graphical Method (for functions of one variable)

For functions of a single variable, a simple graphical approach can determine injectivity. A function is injective if it passes the horizontal line test: no horizontal line intersects the graph more than once.

Example: The graph of f(x) = eˣ passes the horizontal line test, indicating injectivity. Conversely, the graph of g(x) = x² fails the horizontal line test (for all real numbers), confirming it's not injective.

Method 4: Using Calculus (for differentiable functions)

For differentiable functions, we can analyze the derivative. If the derivative is always positive or always negative over the entire domain, the function is strictly monotonic (always increasing or always decreasing) and therefore injective.

Example: The derivative of f(x) = eˣ is f'(x) = eˣ, which is always positive. Therefore, f(x) = eˣ is injective.

Conclusion

Proving injectivity requires a rigorous approach, but understanding the definition and employing the appropriate method—whether direct proof, contraposition, graphical analysis, or calculus—will allow you to effectively demonstrate whether a function maps each element in the domain to a unique element in the codomain. Remember to always carefully consider the domain of the function, as it significantly impacts whether the function is injective. Practice with various examples to build your understanding and proficiency.