How Do You Prove A Function Is One To One

Unlocking the secrets of mathematical functions, understanding when a function is one-to-one—also known as injective—is vital. This article delves into the methods and concepts needed to rigorously prove that a function possesses this unique property. Whether you're a student grappling with calculus, a seasoned mathematician, or simply someone curious about the nature of functions, this comprehensive guide will equip you with the tools to tackle one-to-one function proofs with confidence.

Understanding One-to-One Functions

A function f from a set A to a set B is one-to-one if each element of B is associated with at most one element of A. In simpler terms, a function is one-to-one if different elements in its domain always map to different elements in its range. Mathematically, this can be expressed as:

If f(x₁) = f(x₂), then x₁ = x₂ for all x₁ and x₂ in the domain of f.

This definition is the cornerstone of proving that a function is one-to-one. Understanding this fundamental concept is key to tackling the proofs that follow. A function that is one-to-one passes the horizontal line test, meaning that no horizontal line intersects the graph of the function more than once.

Methods to Prove a Function is One-to-One

There are several methods to prove that a function is one-to-one. Each has its advantages and is suitable for different types of functions. Here are the primary methods:

1. Direct Proof: This involves directly applying the definition of a one-to-one function. You start by assuming that f(x₁) = f(x₂) and then algebraically show that x₁ = x₂.
2. Proof by Contradiction: Assume that f(x₁) = f(x₂) but x₁ ≠ x₂. Then, show that this assumption leads to a contradiction, thus proving the original statement.
3. Proof by Contrapositive: Instead of proving that "if f(x₁) = f(x₂), then x₁ = x₂", prove the contrapositive statement: "if x₁ ≠ x₂, then f(x₁) ≠ f(x₂)".
4. Using Derivatives (Calculus): If the function is differentiable, you can show that its derivative is either always positive or always negative on its domain. This implies that the function is strictly increasing or strictly decreasing, which guarantees that it is one-to-one.

Let's explore each of these methods with examples.

1. Direct Proof

The direct proof method is the most straightforward approach. It directly applies the definition of a one-to-one function.

Steps:

• Assume f(x₁) = f(x₂).
• Use algebraic manipulations to show that x₁ = x₂.

Example 1:

Prove that the function f(x) = 3x + 5 is one-to-one for all real numbers.

Proof:

Assume f(x₁) = f(x₂). This means:

3x₁ + 5 = 3x₂ + 5

Subtract 5 from both sides:

3x₁ = 3x₂

Divide both sides by 3:

x₁ = x₂

Since f(x₁) = f(x₂) implies x₁ = x₂, the function f(x) = 3x + 5 is one-to-one.

Example 2:

Prove that the function g(x) = x³ is one-to-one for all real numbers.

Proof:

Assume g(x₁) = g(x₂). This means:

x₁³ = x₂³

Take the cube root of both sides:

∛(x₁³) = ∛(x₂³)

x₁ = x₂

Since g(x₁) = g(x₂) implies x₁ = x₂, the function g(x) = x³ is one-to-one.

Example 3:

Prove that the function h(x) = (x - 2) / (x + 3) is one-to-one for x ≠ -3.

Proof:

Assume h(x₁) = h(x₂). This means:

(x₁ - 2) / (x₁ + 3) = (x₂ - 2) / (x₂ + 3)

Cross-multiply:

(x₁ - 2)(x₂ + 3) = (x₂ - 2)(x₁ + 3)

Expand both sides:

x₁x₂ + 3x₁ - 2x₂ - 6 = x₁x₂ + 3x₂ - 2x₁ - 6

Simplify:

3x₁ - 2x₂ = 3x₂ - 2x₁

Add 2x₂ and 2x₁ to both sides:

3x₁ + 2x₁ = 3x₂ + 2x₂

5x₁ = 5x₂

Divide by 5:

x₁ = x₂

Since h(x₁) = h(x₂) implies x₁ = x₂, the function h(x) = (x - 2) / (x + 3) is one-to-one.

2. Proof by Contradiction

Proof by contradiction involves assuming the opposite of what you want to prove and showing that this assumption leads to a contradiction.

Steps:

• Assume f(x₁) = f(x₂) and x₁ ≠ x₂.
• Show that this assumption leads to a contradiction.

Example 1:

Prove that the function f(x) = 3x + 5 is one-to-one for all real numbers using proof by contradiction.

Proof:

Assume f(x₁) = f(x₂) and x₁ ≠ x₂. This means:

3x₁ + 5 = 3x₂ + 5

Subtract 5 from both sides:

3x₁ = 3x₂

Divide both sides by 3:

x₁ = x₂

This contradicts our assumption that x₁ ≠ x₂. Therefore, the function f(x) = 3x + 5 is one-to-one.

Example 2:

Prove that the function g(x) = x³ is one-to-one for all real numbers using proof by contradiction.

Proof:

Assume g(x₁) = g(x₂) and x₁ ≠ x₂. This means:

x₁³ = x₂³

Take the cube root of both sides:

∛(x₁³) = ∛(x₂³)

x₁ = x₂

This contradicts our assumption that x₁ ≠ x₂. Therefore, the function g(x) = x³ is one-to-one.

While these examples are quite simple and the direct proof method might seem more straightforward, proof by contradiction can be particularly useful when dealing with more complex functions or when direct proof is difficult to apply.

3. Proof by Contrapositive

Proof by contrapositive involves proving the statement "if x₁ ≠ x₂, then f(x₁) ≠ f(x₂)". This is logically equivalent to the original statement "if f(x₁) = f(x₂), then x₁ = x₂".

Steps:

• Assume x₁ ≠ x₂.
• Show that f(x₁) ≠ f(x₂).

Example 1:

Prove that the function f(x) = 3x + 5 is one-to-one for all real numbers using proof by contrapositive.

Proof:

Assume x₁ ≠ x₂. Multiply both sides by 3:

3x₁ ≠ 3x₂

Add 5 to both sides:

3x₁ + 5 ≠ 3x₂ + 5

This means:

f(x₁) ≠ f(x₂)

Since x₁ ≠ x₂ implies f(x₁) ≠ f(x₂), the function f(x) = 3x + 5 is one-to-one.

Example 2:

Prove that the function g(x) = x³ is one-to-one for all real numbers using proof by contrapositive.

Proof:

Assume x₁ ≠ x₂. Cube both sides:

x₁³ ≠ x₂³

This means:

g(x₁) ≠ g(x₂)

Since x₁ ≠ x₂ implies g(x₁) ≠ g(x₂), the function g(x) = x³ is one-to-one.

Example 3:

Prove that the function h(x) = (x - 2) / (x + 3) is one-to-one for x ≠ -3 using proof by contrapositive.

Proof:

Assume x₁ ≠ x₂. Consider h(x₁) = (x₁ - 2) / (x₁ + 3) and h(x₂) = (x₂ - 2) / (x₂ + 3). We want to show that h(x₁) ≠ h(x₂).

Let's assume, for the sake of contradiction, that h(x₁) = h(x₂). Then:

(x₁ - 2) / (x₁ + 3) = (x₂ - 2) / (x₂ + 3)

Cross-multiply:

(x₁ - 2)(x₂ + 3) = (x₂ - 2)(x₁ + 3)

Expand both sides:

x₁x₂ + 3x₁ - 2x₂ - 6 = x₁x₂ + 3x₂ - 2x₁ - 6

Simplify:

3x₁ - 2x₂ = 3x₂ - 2x₁

Add 2x₂ and 2x₁ to both sides:

5x₁ = 5x₂

Divide by 5:

x₁ = x₂

This contradicts our assumption that x₁ ≠ x₂. Therefore, h(x₁) ≠ h(x₂), and the function h(x) = (x - 2) / (x + 3) is one-to-one.

4. Using Derivatives (Calculus)

If the function is differentiable, calculus provides a powerful tool for proving that it is one-to-one.

Theorem:

If a function f(x) is differentiable on an interval I, and f'(x) > 0 for all x in I or f'(x) < 0 for all x in I, then f(x) is one-to-one on I.

In other words, if the derivative of the function is strictly positive or strictly negative on an interval, the function is strictly increasing or strictly decreasing, respectively, and thus one-to-one.

Steps:

• Find the derivative f'(x) of the function f(x).
• Show that f'(x) > 0 for all x in the domain (strictly increasing) or f'(x) < 0 for all x in the domain (strictly decreasing).

Example 1:

Prove that the function f(x) = eˣ is one-to-one for all real numbers.

Proof:

Find the derivative of f(x):

f'(x) = eˣ

Since eˣ > 0 for all real numbers x, the function f(x) = eˣ is strictly increasing and thus one-to-one.

Example 2:

Prove that the function f(x) = x³ is one-to-one for all real numbers.

Proof:

Find the derivative of f(x):

f'(x) = 3x²

Since 3x² ≥ 0 for all real numbers x, and 3x² = 0 only at x = 0, the function is strictly increasing for x < 0 and x > 0. Therefore, f(x) = x³ is one-to-one.

Example 3:

Prove that the function f(x) = -x⁵ + 2 is one-to-one for all real numbers.

Proof:

Find the derivative of f(x):

f'(x) = -5x⁴

Since -5x⁴ ≤ 0 for all real numbers x, and -5x⁴ = 0 only at x = 0, the function is strictly decreasing for x < 0 and x > 0. Therefore, f(x) = -x⁵ + 2 is one-to-one.

Common Mistakes to Avoid

When proving that a function is one-to-one, it's easy to make mistakes. Here are some common pitfalls to avoid:

1. Assuming what you need to prove: A common mistake is to start with an assumption that is equivalent to the conclusion you want to reach.
2. Algebraic errors: Double-check all algebraic manipulations to avoid errors that can invalidate your proof.
3. Incorrectly applying the definition: Make sure to correctly apply the definition of a one-to-one function in your proofs.
4. Not considering the domain: Always consider the domain of the function, as it can affect whether the function is one-to-one.
5. Confusing one-to-one with onto: A function can be one-to-one, onto (surjective), both (bijective), or neither. Make sure you're proving the correct property.
6. Assuming differentiability: When using derivatives, ensure that the function is indeed differentiable on the interval you are considering.
7. Not providing a clear conclusion: Always state your conclusion clearly after completing your proof.

Examples of Functions That Are Not One-to-One

Understanding functions that are not one-to-one can further clarify the concept. Here are some examples:

1. Quadratic Functions: f(x) = x² is not one-to-one because f(2) = 4 and f(-2) = 4.
2. Trigonometric Functions: f(x) = sin(x) is not one-to-one because f(0) = 0 and f(π) = 0.
3. Absolute Value Functions: f(x) = |x| is not one-to-one because f(3) = 3 and f(-3) = 3.

For these functions, it is easy to find two different values x₁ and x₂ such that f(x₁) = f(x₂), thus violating the definition of a one-to-one function.

Practice Problems

To solidify your understanding, here are some practice problems:

1. Prove that f(x) = 5x - 2 is one-to-one using direct proof.
2. Prove that g(x) = x⁵ is one-to-one using proof by contrapositive.
3. Prove that h(x) = (2x + 1) / (x - 3) is one-to-one for x ≠ 3 using direct proof.
4. Prove that f(x) = √x is one-to-one for x ≥ 0 using the derivative test.
5. Show that f(x) = x⁴ is not one-to-one.

Conclusion

Proving that a function is one-to-one is a fundamental skill in mathematics. By mastering the direct proof, proof by contradiction, proof by contrapositive, and the use of derivatives, you can confidently tackle a wide range of functions. Remember to avoid common mistakes, understand the importance of the domain, and practice regularly to strengthen your understanding. With dedication and practice, you'll become proficient in determining whether a function is one-to-one and providing rigorous proofs to support your conclusions.