How To Prove That A Function Is One To One

In mathematics, proving that a function is one-to-one, also known as injective, is a fundamental concept with applications in various fields. A function is one-to-one if each element of the range is associated with a unique element of the domain. In simpler terms, if f(x₁) = f(x₂), then x₁ must be equal to x₂. This article provides a comprehensive guide on how to prove that a function is one-to-one, covering different methods, examples, and advanced techniques.

Understanding One-to-One Functions

Before diving into the methods of proving a function is one-to-one, it's crucial to understand the concept thoroughly. A function f: A → B is one-to-one (injective) if for every b ∈ B, there is at most one a ∈ A such that f(a) = b. This means no two different elements in the domain A map to the same element in the range B.

Key Definitions

• Function: A relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output.
• Domain (A): The set of all possible input values (x) for a function.
• Range (B): The set of all output values (f(x)) that result from using all possible input values.
• One-to-One (Injective) Function: A function where each element of the range corresponds to exactly one element of the domain.
• Not One-to-One (Non-Injective) Function: A function where at least one element of the range corresponds to more than one element of the domain.

Why is Injectivity Important?

One-to-one functions are crucial in many areas of mathematics and its applications:

• Inverse Functions: A function has an inverse if and only if it is one-to-one. Inverse functions are essential for solving equations and understanding transformations.
• Cryptography: Injective functions are used in encryption algorithms to ensure that each plaintext message maps to a unique ciphertext message.
• Data Compression: Injective functions are used to compress data without losing information.
• Mathematical Proofs: Injectivity is a fundamental property used in various mathematical proofs, especially in set theory and analysis.

Methods to Prove a Function is One-to-One

There are several methods to prove that a function is one-to-one. The choice of method depends on the nature of the function. Here are the common techniques:

1. Direct Proof:

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

2. Proof by Contradiction:

   • Assume f(x₁) = f(x₂) and x₁ ≠ x₂.
   • Derive a contradiction.

3. Using the Definition of Monotonicity:

   • Show that the function is strictly increasing or strictly decreasing over its entire domain.

4. Horizontal Line Test:

   • Graph the function.
   • If any horizontal line intersects the graph at most once, the function is one-to-one.

5. Derivative Test (Calculus):

   • Compute the derivative f'(x).
   • If f'(x) > 0 for all x in the domain or f'(x) < 0 for all x in the domain, the function is one-to-one.

1. Direct Proof Method

The direct proof method is the most straightforward way to prove a function is one-to-one. It involves starting with the assumption that f(x₁) = f(x₂) and then algebraically manipulating the equation to show that x₁ = x₂.

Steps:

1. Assume f(x₁) = f(x₂) for arbitrary x₁ and x₂ in the domain of f.
2. Simplify the equation f(x₁) = f(x₂) using algebraic manipulations.
3. Show that the simplified equation leads to x₁ = x₂.
4. Conclude that since f(x₁) = f(x₂) implies x₁ = x₂, the function f is one-to-one.

Example:

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

1. Assume: f(x₁) = f(x₂)
2. Write the equation: 3x₁ + 5 = 3x₂ + 5
3. Simplify:
   • Subtract 5 from both sides: 3x₁ = 3x₂
   • Divide both sides by 3: x₁ = x₂
4. Conclude: Since f(x₁) = f(x₂) implies x₁ = x₂, the function f(x) = 3x + 5 is one-to-one.

2. Proof by Contradiction Method

The proof by contradiction method involves assuming that the function is not one-to-one and then deriving a contradiction. This contradiction shows that the initial assumption must be false, proving that the function is one-to-one.

Steps:

1. Assume that f(x₁) = f(x₂) and x₁ ≠ x₂ for some x₁ and x₂ in the domain of f.
2. Simplify the equation f(x₁) = f(x₂) using algebraic manipulations.
3. Derive a contradiction, i.e., show that the assumption leads to a statement that is logically impossible or contradicts a known fact.
4. Conclude that since the assumption leads to a contradiction, it must be false. Therefore, f(x₁) = f(x₂) implies x₁ = x₂, and the function f is one-to-one.

Example:

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

1. Assume: f(x₁) = f(x₂) and x₁ ≠ x₂
2. Write the equation: x₁³ = x₂³
3. Simplify:
   • Take the cube root of both sides: x₁ = x₂
4. Derive a contradiction: This contradicts the initial assumption that x₁ ≠ x₂.
5. Conclude: Since the assumption x₁ ≠ x₂ leads to a contradiction, it must be false. Therefore, f(x₁) = f(x₂) implies x₁ = x₂, and the function f(x) = x³ is one-to-one.

3. Using the Definition of Monotonicity

A function is strictly increasing if for all x₁ and x₂ in the domain, if x₁ < x₂, then f(x₁) < f(x₂). Similarly, a function is strictly decreasing if for all x₁ and x₂ in the domain, if x₁ < x₂, then f(x₁) > f(x₂). If a function is either strictly increasing or strictly decreasing over its entire domain, it is one-to-one.

Steps:

1. Show that the function is either strictly increasing or strictly decreasing.
2. Conclude that since the function is monotonic, it is one-to-one.

Example:

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

1. Show: f(x) = eˣ is strictly increasing.
   • Let x₁ < x₂.
   • Then eˣ¹ < eˣ².
   • Thus, f(x₁) < f(x₂).
2. Conclude: Since f(x) = eˣ is strictly increasing, it is one-to-one.

4. Horizontal Line Test

The horizontal line test is a graphical method to determine if a function is one-to-one. If any horizontal line intersects the graph of the function at most once, the function is one-to-one.

Steps:

1. Graph the function f(x).
2. Draw horizontal lines across the graph.
3. Check if any horizontal line intersects the graph more than once.
4. Conclude:
   • If no horizontal line intersects the graph more than once, the function is one-to-one.
   • If any horizontal line intersects the graph more than once, the function is not one-to-one.

Example:

Consider the function f(x) = x².

1. Graph: The graph of f(x) = x² is a parabola.
2. Draw: Horizontal lines across the graph.
3. Check: Horizontal lines above the x-axis intersect the graph twice.
4. Conclude: Since horizontal lines intersect the graph more than once, the function f(x) = x² is not one-to-one.

5. Derivative Test (Calculus)

The derivative test involves using calculus to determine if a function is one-to-one. If the derivative f'(x) is either always positive or always negative over the entire domain of the function, then the function is one-to-one.

Steps:

1. Compute the derivative f'(x) of the function f(x).
2. Determine the sign of f'(x) over the domain of f.
3. Conclude:
   • If f'(x) > 0 for all x in the domain, the function is strictly increasing and one-to-one.
   • If f'(x) < 0 for all x in the domain, the function is strictly decreasing and one-to-one.
   • If f'(x) changes sign, the function is not one-to-one.

Example:

Prove that the function f(x) = x³ + 2x + 1 is one-to-one.

1. Compute: f'(x) = 3x² + 2
2. Determine: Since x² ≥ 0 for all x, then 3x² ≥ 0, and 3x² + 2 ≥ 2. Therefore, f'(x) > 0 for all x.
3. Conclude: Since f'(x) > 0 for all x, the function f(x) = x³ + 2x + 1 is strictly increasing and one-to-one.

Advanced Techniques and Special Cases

In addition to the basic methods, there are advanced techniques and special cases to consider when proving that a function is one-to-one.

Composition of Functions

If f and g are both one-to-one functions, then their composition f ∘ g is also one-to-one.

Proof:

1. Assume: (f ∘ g)(x₁) = (f ∘ g)(x₂)
2. Write the equation: f(g(x₁)) = f(g(x₂))
3. Since f is one-to-one: g(x₁) = g(x₂)
4. Since g is one-to-one: x₁ = x₂
5. Conclude: Therefore, (f ∘ g)(x₁) = (f ∘ g)(x₂) implies x₁ = x₂, and the composition f ∘ g is one-to-one.

Functions with Restricted Domains

Sometimes a function may not be one-to-one over its entire natural domain, but it can be one-to-one if the domain is restricted.

Example:

The function f(x) = x² is not one-to-one over its entire domain (all real numbers) because f(-x) = f(x). However, if we restrict the domain to x ≥ 0, then the function becomes one-to-one.

Proof:

1. Assume: f(x₁) = f(x₂) for x₁, x₂ ≥ 0
2. Write the equation: x₁² = x₂²
3. Simplify:
   • Take the square root of both sides: x₁ = ±x₂
   • Since x₁, x₂ ≥ 0, we have x₁ = x₂
4. Conclude: Therefore, f(x₁) = f(x₂) implies x₁ = x₂, and the function f(x) = x² is one-to-one on the domain x ≥ 0.

Piecewise Functions

For piecewise functions, you need to check the injectivity of each piece separately and also ensure that the pieces connect in a way that preserves injectivity.

Example:

Consider the piecewise function:

f(x) = {
  x,    if x < 0
  x²,   if x ≥ 0
}

This function is not one-to-one because for x ≥ 0, f(x) = x², which is not one-to-one as shown earlier.

However, if we modify the function to:

f(x) = {
  x,    if x < 0
  √x,   if x ≥ 0
}

Then the function is one-to-one.

Proof:

1. For x < 0: f(x) = x is one-to-one.
2. For x ≥ 0: f(x) = √x is one-to-one.
3. Check the connection: The range of f(x) = x for x < 0 is (−∞, 0), and the range of f(x) = √x for x ≥ 0 is [0, ∞). These ranges do not overlap except at 0, and f(0) = 0.

Thus, the function is one-to-one.

Common Mistakes to Avoid

When proving that a function is one-to-one, it's essential to avoid common mistakes:

• Assuming the Conclusion: Do not start by assuming that x₁ = x₂. The goal is to prove this equality.
• Incorrect Algebraic Manipulations: Ensure that each algebraic step is valid. For example, taking the square root of both sides of an equation requires considering both positive and negative roots unless there is a constraint on the domain.
• Ignoring Domain Restrictions: Always consider the domain of the function. A function may be one-to-one on a restricted domain but not on its entire natural domain.
• Confusing One-to-One with Onto: A function is one-to-one (injective) if it maps distinct elements of its domain to distinct elements of its codomain. A function is onto (surjective) if its range is equal to its codomain. These are different properties.
• Using Examples as Proof: Providing examples where f(x₁) = f(x₂) implies x₁ = x₂ does not constitute a proof. A proof must be general and cover all possible values in the domain.

Practical Examples and Exercises

To solidify your understanding, here are some practical examples and exercises:

Example 1:

Prove that the function f(x) = (x - 2) / (x + 3), where x ≠ -3, is one-to-one.

Solution:

1. Assume: f(x₁) = f(x₂)
2. Write the equation: (x₁ - 2) / (x₁ + 3) = (x₂ - 2) / (x₂ + 3)
3. Simplify:
   • Cross-multiply: (x₁ - 2)(x₂ + 3) = (x₂ - 2)(x₁ + 3)
   • Expand: x₁x₂ + 3x₁ - 2x₂ - 6 = x₁x₂ + 3x₂ - 2x₁ - 6
   • Simplify: 3x₁ - 2x₂ = 3x₂ - 2x₁
   • Rearrange: 5x₁ = 5x₂
   • Divide by 5: x₁ = x₂
4. Conclude: Since f(x₁) = f(x₂) implies x₁ = x₂, the function f(x) = (x - 2) / (x + 3) is one-to-one.

Exercise 1:

Prove that the function f(x) = 4x - 7 is one-to-one.

Exercise 2:

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

Exercise 3:

Determine whether the function f(x) = |x| is one-to-one. If not, can you restrict the domain to make it one-to-one?

Exercise 4:

Use the derivative test to show that f(x) = x⁵ + 3x is one-to-one.

Conclusion

Proving that a function is one-to-one is a fundamental skill in mathematics with wide-ranging applications. This article has provided a comprehensive guide to various methods for proving injectivity, including direct proof, proof by contradiction, using monotonicity, the horizontal line test, and the derivative test. Additionally, advanced techniques such as composition of functions and handling functions with restricted domains or piecewise definitions were discussed. By understanding these methods and avoiding common mistakes, you can effectively prove whether a function is one-to-one. Regular practice with examples and exercises will further enhance your proficiency in this area.