Proving A Function Is One To One

Unlocking the secrets to show a function is one-to-one, also known as injective, is crucial in mathematics. This property ensures that each element in the range corresponds to a unique element in the domain, setting the stage for invertible functions and a deeper understanding of mathematical relationships.

Understanding One-to-One Functions

A function f is considered one-to-one if different elements in its domain always map to different elements in its range. More formally, if f(x₁) = f(x₂), then x₁ = x₂. This definition is the key to proving that a function is one-to-one. Intuitively, a one-to-one function never assigns the same value to two different inputs.

Why One-to-One Matters

The one-to-one property is fundamental for several reasons:

• Invertibility: A function must be one-to-one to have an inverse. The inverse function undoes the action of the original function, and this is only possible if each output corresponds to a unique input.
• Uniqueness: In various mathematical models, ensuring uniqueness is critical. One-to-one functions guarantee that each outcome is uniquely associated with its cause.
• Applications: One-to-one functions are used in cryptography, data compression, and many other fields where unique mappings are essential.

Methods to Prove a Function Is One-to-One

There are several methods to prove that a function is one-to-one. Each method has its strengths and is suitable for different types of functions. The main methods include:

1. Direct Proof: Using the definition f(x₁) = f(x₂) implies x₁ = x₂.
2. Proof by Contrapositive: Showing that x₁ ≠ x₂ implies f(x₁) ≠ f(x₂).
3. Using Derivatives (for differentiable functions): Showing that the function is strictly increasing or strictly decreasing.
4. Graphical Method: Using the horizontal line test.
5. Proof by Contradiction: Assuming the function is not one-to-one and deriving a contradiction.

Let's explore each method in detail with examples.

1. Direct Proof

The direct proof method involves directly applying the definition of a one-to-one function.

Steps:

1. Assume f(x₁) = f(x₂) for arbitrary x₁ and x₂ in the domain of f.
2. Use algebraic manipulations to show that x₁ = x₂.
3. Conclude that f is one-to-one.

Example:

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

Proof:

Assume f(x₁) = f(x₂) for some x₁ and x₂. Then,

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.

Another Example:

Prove that the function f(x) = x³ is one-to-one.

Proof:

Assume f(x₁) = f(x₂) for some x₁ and x₂. Then,

x₁³ = x₂³

Take the cube root of both sides:

∛(x₁³) = ∛(x₂³)

x₁ = x₂

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

2. Proof by Contrapositive

The contrapositive of the statement "If A, then B" is "If not B, then not A." In the context of one-to-one functions, the contrapositive of "f(x₁) = f(x₂) implies x₁ = x₂" is "x₁ ≠ x₂ implies f(x₁) ≠ f(x₂)".

Steps:

1. Assume x₁ ≠ x₂ for arbitrary x₁ and x₂ in the domain of f.
2. Use algebraic manipulations to show that f(x₁) ≠ f(x₂).
3. Conclude that f is one-to-one.

Example:

Prove that the function f(x) = 2x - 3 is one-to-one using the contrapositive method.

Proof:

Assume x₁ ≠ x₂. Then,

Multiply both sides by 2:

2x₁ ≠ 2x₂

Subtract 3 from both sides:

2x₁ - 3 ≠ 2x₂ - 3

Thus, f(x₁) ≠ f(x₂).

Since x₁ ≠ x₂ implies f(x₁) ≠ f(x₂), the function f(x) = 2x - 3 is one-to-one.

Another Example:

Prove that the function f(x) = x³ + 1 is one-to-one using the contrapositive method.

Proof:

Assume x₁ ≠ x₂. Then,

We want to show that f(x₁) ≠ f(x₂), i.e., x₁³ + 1 ≠ x₂³ + 1.

Since x₁ ≠ x₂, we know that x₁³ ≠ x₂³ (as the cube function is one-to-one).

Adding 1 to both sides:

x₁³ + 1 ≠ x₂³ + 1

Thus, f(x₁) ≠ f(x₂).

Since x₁ ≠ x₂ implies f(x₁) ≠ f(x₂), the function f(x) = x³ + 1 is one-to-one.

3. Using Derivatives (for Differentiable Functions)

If a function is differentiable over its entire domain, we can use its derivative to determine if it is strictly increasing or strictly decreasing. A strictly increasing or strictly decreasing function is always one-to-one.

Steps:

1. Find the derivative f'(x) of the function f(x).
2. 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).
3. Conclude that f is one-to-one.

Example:

Prove that the function f(x) = eˣ is one-to-one using derivatives.

Proof:

Find the derivative of f(x) = eˣ:

f'(x) = eˣ

Since eˣ > 0 for all x in the domain (which is all real numbers), f'(x) > 0 for all x.

Thus, f(x) = eˣ is strictly increasing, and therefore, it is one-to-one.

Another Example:

Prove that the function f(x) = -x³ + 2 is one-to-one using derivatives.

Proof:

Find the derivative of f(x) = -x³ + 2:

f'(x) = -3x²

Since -3x² ≤ 0 for all x, the function is non-increasing. However, it is strictly decreasing for all x ≠ 0. Thus, f'(x) < 0 for all x ≠ 0.

However, for a more rigorous approach, we must ensure f'(x) is not zero on any interval. Since f'(x) = 0 only at x = 0, the function is strictly decreasing on any interval not containing 0. Therefore, we can conclude that the function f(x) = -x³ + 2 is one-to-one.

4. Graphical Method (Horizontal Line Test)

The horizontal line test is a visual method to determine if a function is one-to-one.

Steps:

1. Graph the function f(x).
2. Draw horizontal lines across the graph.
3. If no horizontal line intersects the graph more than once, then the function is one-to-one.

Example:

Consider the function f(x) = x². If we graph this function, we see that a horizontal line (e.g., y = 4) intersects the graph at two points (x = 2 and x = -2). Therefore, f(x) = x² is not one-to-one.

Another Example:

Consider the function f(x) = x³. If we graph this function, any horizontal line will intersect the graph at most once. Therefore, f(x) = x³ is one-to-one.

5. 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:

1. Assume that f is not one-to-one. This means there exist x₁ ≠ x₂ such that f(x₁) = f(x₂).
2. Use this assumption to derive a contradiction.
3. Conclude that the original assumption (that f is not one-to-one) is false, and therefore f is one-to-one.

Example:

Prove that the function f(x) = x + 7 is one-to-one using proof by contradiction.

Proof:

Assume that f is not one-to-one. Then there exist x₁ ≠ x₂ such that f(x₁) = f(x₂).

So, x₁ + 7 = x₂ + 7.

Subtracting 7 from both sides, we get:

x₁ = x₂

This contradicts our assumption that x₁ ≠ x₂. Therefore, our assumption that f is not one-to-one is false.

Thus, the function f(x) = x + 7 is one-to-one.

Another Example:

Prove that the function f(x) = 1/x for x ≠ 0 is one-to-one using proof by contradiction.

Proof:

Assume that f is not one-to-one. Then there exist x₁ ≠ x₂ such that f(x₁) = f(x₂).

So, 1/x₁ = 1/x₂.

Multiplying both sides by x₁x₂, we get:

x₂ = x₁

This contradicts our assumption that x₁ ≠ x₂. Therefore, our assumption that f is not one-to-one is false.

Thus, the function f(x) = 1/x is one-to-one.

Examples and Non-Examples

To solidify your understanding, let's look at some more examples and non-examples of one-to-one functions.

One-to-One Functions:

• f(x) = x (identity function)
• f(x) = mx + b, where m ≠ 0 (linear function with non-zero slope)
• f(x) = √x for x ≥ 0 (square root function)
• f(x) = ln(x) for x > 0 (natural logarithm function)
• f(x) = tan⁻¹(x) (inverse tangent function)

Non-One-to-One Functions:

• f(x) = x² (quadratic function)
• f(x) = |x| (absolute value function)
• f(x) = sin(x) (sine function)
• f(x) = cos(x) (cosine function)
• f(x) = x⁴ + 2x² + 1 (even-powered polynomial with symmetry)

Practical Tips for Proving One-to-One

• Choose the Right Method: Consider the type of function you are dealing with. Derivatives are useful for differentiable functions, while direct proof or contrapositive might be better for algebraic functions.
• Be Careful with Domains: Always consider the domain of the function. A function might be one-to-one on a restricted domain but not on its entire domain. For example, f(x) = x² is not one-to-one on the entire real line but is one-to-one for x ≥ 0.
• Algebraic Manipulations: Be meticulous with your algebraic manipulations to avoid errors.
• Understand the Definition: Always refer back to the definition of a one-to-one function to guide your proof.

Advanced Topics

Injectivity and Surjectivity

A function is injective if it is one-to-one. A function is surjective (or onto) if every element in the codomain is the image of at least one element in the domain. A function that is both injective and surjective is called bijective.

Bijections and Invertibility

A function is invertible if and only if it is bijective. The inverse function, denoted as f⁻¹, reverses the mapping of f. If f(x) = y, then f⁻¹(y) = x. The existence of an inverse function is a powerful tool in mathematics.

Composition of One-to-One Functions

If f and g are both one-to-one functions, then their composition f(g(x)) is also one-to-one. This property is useful when dealing with complex functions that can be decomposed into simpler one-to-one functions.

Common Mistakes to Avoid

• Assuming One-to-One: Do not assume that a function is one-to-one without proof. Always provide a rigorous argument.
• Incorrect Algebraic Manipulations: Double-check your algebraic steps to avoid errors.
• Ignoring the Domain: Always consider the domain of the function when determining if it is one-to-one.
• Misapplying the Horizontal Line Test: Ensure you understand how to apply the horizontal line test correctly.

Conclusion

Proving that a function is one-to-one is a fundamental skill in mathematics. By understanding the definition of one-to-one functions and mastering the various proof techniques, you can confidently tackle a wide range of problems. Whether using direct proof, contrapositive, derivatives, graphical methods, or proof by contradiction, the key is to approach each problem with a clear understanding of the underlying principles and a meticulous approach to the proof. With practice, you'll become adept at recognizing and proving the one-to-one property of functions, opening doors to more advanced mathematical concepts and applications.