The One To One Function H Is Defined Below

Let's delve into the fascinating world of functions, specifically focusing on one-to-one functions and how to determine if a function, denoted as h in this context, meets the criteria. Understanding one-to-one functions is crucial in various areas of mathematics, computer science, and beyond. This article aims to provide a comprehensive explanation, illustrated with examples and methods for verification.

Understanding One-to-One Functions: The Basics

A one-to-one function, also known as an injective function, is a function where each element of the range (the set of output values) is associated with exactly one element of the domain (the set of input values). In simpler terms, no two different inputs produce the same output. This characteristic is the defining feature of a one-to-one function and distinguishes it from other types of functions.

Formally, a function h is one-to-one if and only if for any two elements x₁ and x₂ in the domain of h, if h(x₁) = h(x₂), then x₁ = x₂. Conversely, if x₁ ≠ x₂, then h(x₁) ≠ h(x₂).

Let's break down this definition:

• Domain: The set of all possible input values for the function (often represented by x).
• Range: The set of all actual output values produced by the function (often represented by h(x) or y).
• One-to-one: Each y value in the range corresponds to only one x value in the domain.

Methods to Determine if a Function h is One-to-One

Several methods can be used to determine whether a function h is one-to-one. These methods include:

1. Horizontal Line Test (Graphical Method): This method applies when the function is represented graphically.
2. Algebraic Method: This method involves using the formal definition of a one-to-one function.
3. Derivative Test (Calculus Method): This method utilizes calculus concepts (specifically derivatives) and is applicable to differentiable functions.

Let's explore each method in detail.

1. Horizontal Line Test (Graphical Method)

The horizontal line test provides a visual way to determine if a function is one-to-one. If any horizontal line intersects the graph of the function h at more than one point, then the function is not one-to-one. Conversely, if every horizontal line intersects the graph at most once, then the function is one-to-one.

How to apply the Horizontal Line Test:

1. Graph the function h(x): Accurately plot the graph of the function on a coordinate plane.
2. Draw horizontal lines: Imagine (or actually draw) horizontal lines across the graph at various y values.
3. Check for intersections: Observe how many times each horizontal line intersects the graph of the function.
4. Interpret the results:
   • If any horizontal line intersects the graph at more than one point, the function is not one-to-one.
   • If every horizontal line intersects the graph at most one point, the function is one-to-one.

Examples:

• Example 1: h(x) = x² (Parabola)

  If you graph h(x) = x², you'll see a parabola. A horizontal line, for instance, y = 4, will intersect the graph at x = 2 and x = -2. Since the horizontal line intersects the graph at two points, the function h(x) = x² is not one-to-one.

• Example 2: h(x) = x³ (Cubic Function)

  The graph of h(x) = x³ is a cubic function that increases steadily. Any horizontal line will intersect the graph at only one point. Therefore, the function h(x) = x³ is one-to-one.

• Example 3: h(x) = sin(x) (Sine Function)

  The graph of h(x) = sin(x) is a wave that oscillates between -1 and 1. Many horizontal lines will intersect the graph at infinitely many points. Therefore, the function h(x) = sin(x) is not one-to-one.

Advantages of the Horizontal Line Test:

• Visual: It provides a clear and intuitive visual representation of the function's behavior.
• Quick: It's often a quick way to determine if a function is not one-to-one.

Disadvantages of the Horizontal Line Test:

• Requires a graph: It necessitates having the graph of the function, which might not always be readily available.
• Subjectivity: Interpretation can sometimes be subjective, especially with complex graphs.

2. Algebraic Method

The algebraic method relies on the formal definition of a one-to-one function. You start by assuming that h(x₁) = h(x₂) and then algebraically manipulate the equation to see if you can prove that x₁ = x₂.

How to apply the Algebraic Method:

1. Assume h(x₁) = h(x₂): Begin by setting the function evaluated at two arbitrary inputs, x₁ and x₂, equal to each other.
2. Algebraically manipulate the equation: Use algebraic techniques to simplify the equation h(x₁) = h(x₂). The goal is to isolate x₁ and x₂.
3. Check if x₁ = x₂: If, after simplification, you can definitively conclude that x₁ = x₂, then the function h is one-to-one. If you cannot reach this conclusion, the function is not one-to-one.

Examples:

• Example 1: h(x) = 2x + 3 (Linear Function)

  1. Assume h(x₁) = h(x₂): 2x₁ + 3 = 2x₂ + 3
  2. Algebraically manipulate:
     • Subtract 3 from both sides: 2x₁ = 2x₂
     • Divide both sides by 2: x₁ = x₂
  3. Conclusion: Since x₁ = x₂, the function h(x) = 2x + 3 is one-to-one.

• Example 2: h(x) = x² (Quadratic Function)

  1. Assume h(x₁) = h(x₂): x₁² = x₂²
  2. Algebraically manipulate:
     • Subtract x₂² from both sides: x₁² - x₂² = 0
     • Factor the difference of squares: (x₁ - x₂)(x₁ + x₂) = 0
     • This implies either x₁ - x₂ = 0 or x₁ + x₂ = 0
     • Therefore, x₁ = x₂ or x₁ = -x₂
  3. Conclusion: Since x₁ can be equal to either x₂ or -x₂, we cannot definitively conclude that x₁ = x₂. Therefore, the function h(x) = x² is not one-to-one.

• Example 3: h(x) = (x - 1) / (x + 2), x ≠ -2 (Rational Function)

  1. Assume h(x₁) = h(x₂): (x₁ - 1) / (x₁ + 2) = (x₂ - 1) / (x₂ + 2)
  2. Algebraically manipulate:
     • Cross-multiply: (x₁ - 1)(x₂ + 2) = (x₂ - 1)(x₁ + 2)
     • Expand both sides: x₁x₂ + 2x₁ - x₂ - 2 = x₁x₂ + 2x₂ - x₁ - 2
     • Simplify: 2x₁ - x₂ = 2x₂ - x₁
     • Combine like terms: 3x₁ = 3x₂
     • Divide both sides by 3: x₁ = x₂
  3. Conclusion: Since x₁ = x₂, the function h(x) = (x - 1) / (x + 2) is one-to-one.

Advantages of the Algebraic Method:

• Rigorous: It provides a mathematically rigorous way to prove whether a function is one-to-one.
• Applicable to many functions: It can be applied to a wide range of function types.

Disadvantages of the Algebraic Method:

• Can be complex: The algebraic manipulations can sometimes be quite challenging, especially for more complex functions.

3. Derivative Test (Calculus Method)

The derivative test uses calculus to determine if a differentiable function is one-to-one. A function h(x) is one-to-one if its derivative, h'(x), is either always strictly positive (h'(x) > 0) or always strictly negative (h'(x) < 0) over its entire domain (excluding points where the derivative is undefined). In other words, the function must be either strictly increasing or strictly decreasing.

How to apply the Derivative Test:

1. Find the derivative h'(x): Calculate the derivative of the function h(x) with respect to x.
2. Analyze the sign of h'(x): Determine the intervals where h'(x) > 0 (function is increasing) and where h'(x) < 0 (function is decreasing).
3. Interpret the results:
   • If h'(x) > 0 for all x in the domain (except possibly at isolated points where h'(x) = 0), or if h'(x) < 0 for all x in the domain (except possibly at isolated points where h'(x) = 0), then the function h(x) is one-to-one.
   • If h'(x) changes sign within the domain, then the function h(x) is not one-to-one.

Examples:

• Example 1: h(x) = x³ (Cubic Function)

  1. Find the derivative: h'(x) = 3x²
  2. Analyze the sign: h'(x) = 3x² is always greater than or equal to 0. It's only equal to 0 at x = 0.
  3. Conclusion: Since h'(x) ≥ 0 for all x, and h'(x) = 0 only at one isolated point, the function h(x) = x³ is one-to-one. (Note: A slightly more nuanced version of this test exists that allows for the derivative to be zero at isolated points.)

• Example 2: h(x) = x² (Quadratic Function)

  1. Find the derivative: h'(x) = 2x
  2. Analyze the sign: h'(x) = 2x is positive for x > 0 and negative for x < 0.
  3. Conclusion: Since h'(x) changes sign, the function h(x) = x² is not one-to-one.

• Example 3: h(x) = eˣ (Exponential Function)

  1. Find the derivative: h'(x) = eˣ
  2. Analyze the sign: h'(x) = eˣ is always positive for all x.
  3. Conclusion: Since h'(x) > 0 for all x, the function h(x) = eˣ is one-to-one.

Advantages of the Derivative Test:

• Applicable to differentiable functions: It provides a powerful method for analyzing differentiable functions.
• Systematic: It offers a systematic approach based on calculus principles.

Disadvantages of the Derivative Test:

• Requires differentiability: It can only be applied to functions that are differentiable.
• Doesn't always apply: A function can be one-to-one but not differentiable everywhere. This test wouldn't apply directly in such cases.

Examples Combining Methods

Let's consider the function h(x) = √x for x ≥ 0.

• Horizontal Line Test: The graph of h(x) = √x starts at the origin and increases gradually. Any horizontal line will intersect the graph at most once. Therefore, based on the Horizontal Line Test, the function appears to be one-to-one.

• Algebraic Method:

  1. Assume h(x₁) = h(x₂): √x₁ = √x₂
  2. Algebraically manipulate:
     • Square both sides: x₁ = x₂
  3. Conclusion: Since x₁ = x₂, the function h(x) = √x is one-to-one.

• Derivative Test:

  1. Find the derivative: h'(x) = 1 / (2√x)
  2. Analyze the sign: h'(x) = 1 / (2√x) is always positive for x > 0. The derivative is undefined at x = 0.
  3. Conclusion: Since h'(x) > 0 for all x > 0, the function h(x) = √x is one-to-one.

All three methods confirm that h(x) = √x for x ≥ 0 is a one-to-one function.

Importance of One-to-One Functions

One-to-one functions are important for several reasons:

• Invertibility: A function has an inverse function if and only if it is one-to-one. The inverse function "undoes" the original function. If a function is not one-to-one, it's impossible to define a unique inverse.
• Mathematical Theorems: Many mathematical theorems and proofs rely on the properties of one-to-one functions.
• Computer Science: In computer science, one-to-one functions are used in cryptography, hashing algorithms, and data compression.
• Data Analysis: In data analysis, one-to-one functions can be used to map data from one domain to another while preserving the relationships between the data points.

Common Mistakes to Avoid

• Assuming all functions are one-to-one: Not all functions are one-to-one. It's crucial to verify using one of the methods described above.
• Misinterpreting the Horizontal Line Test: Make sure to draw horizontal lines across the entire graph to ensure no line intersects at more than one point.
• Incorrect Algebraic Manipulation: Carefully perform the algebraic steps to avoid errors that could lead to an incorrect conclusion.
• Ignoring the Domain: The domain of the function is crucial. A function might be one-to-one over a specific restricted domain but not one-to-one over its entire natural domain. For example, h(x) = x² is not one-to-one over the entire real number line, but it is one-to-one if we restrict the domain to x ≥ 0.

Conclusion

Determining whether a function h is one-to-one is a fundamental concept in mathematics. By understanding the definition of one-to-one functions and applying the horizontal line test, algebraic method, or derivative test, you can effectively determine if a function possesses this important property. Remember to consider the domain of the function and avoid common mistakes in your analysis. The ability to identify one-to-one functions is essential for understanding invertibility, applying mathematical theorems, and solving problems in various fields like computer science and data analysis. Mastering these techniques will greatly enhance your mathematical toolkit.