Which Relation Below Represents A One To One Function

Which Relation Below Represents a One-to-One Function? A Deep Dive into Functions and Their Properties

Understanding functions is fundamental to mathematics and computer science. Within the realm of functions, a particularly important concept is the one-to-one function, also known as an injective function. This article will delve into the definition of a one-to-one function, explore various methods for identifying them, and provide examples and non-examples to solidify your understanding. We'll also touch upon the inverse function, a key characteristic closely linked to one-to-one functions.

Defining a One-to-One Function

A function, at its core, is a relation between a set of inputs (the domain) and a set of possible outputs (the codomain), where each input is associated with exactly one output. However, a one-to-one function imposes a stricter condition: each output is associated with exactly one input. In other words, no two different inputs can produce the same output.

This can be formally expressed as: If f(x₁) = f(x₂), then x₁ = x₂. Conversely, if x₁ ≠ x₂, then f(x₁) ≠ f(x₂).

Let's visualize this concept. Imagine a vending machine. A regular function is like a vending machine where each button (input) dispenses a specific item (output). A one-to-one function is like a vending machine where each item (output) is dispensed by only one button (input). You can't get the same item from two different buttons.

Methods for Identifying One-to-One Functions

Several methods can help you determine whether a given relation is a one-to-one function.

1. The Horizontal Line Test

This is a graphical method applicable when the function is represented visually. If any horizontal line intersects the graph of the function at more than one point, then the function is not one-to-one. If every horizontal line intersects the graph at most once, the function is one-to-one. This test leverages the definition directly; if two points share the same y-coordinate, they violate the one-to-one property.

2. Algebraic Approach

This method involves using the definition directly. Assume f(x₁) = f(x₂) and then manipulate the equation algebraically to show that x₁ must equal x₂. If you can successfully demonstrate this, the function is one-to-one. If you find a scenario where f(x₁) = f(x₂) but x₁ ≠ x₂, the function is not one-to-one.

3. Examining the Function's Behavior

For some functions, understanding their inherent behavior can help determine whether they are one-to-one. For instance, strictly increasing or strictly decreasing functions are always one-to-one. A strictly increasing function means that for any x₁ < x₂, f(x₁) < f(x₂). Similarly, a strictly decreasing function implies that for any x₁ < x₂, f(x₁) > f(x₂).

Examples of One-to-One Functions

Let's examine some examples to illustrate the concept.

• f(x) = x: This is the simplest example. If f(x₁) = f(x₂), then x₁ = x₂. Therefore, it's a one-to-one function. Its graph is a straight line with a slope of 1, and no horizontal line can intersect it more than once.

• f(x) = 2x + 1: Let's use the algebraic approach. Assume f(x₁) = f(x₂). This means 2x₁ + 1 = 2x₂ + 1. Subtracting 1 from both sides and then dividing by 2, we get x₁ = x₂. This confirms it's a one-to-one function.

• f(x) = x³: This function is strictly increasing for all real numbers. Therefore, it's a one-to-one function. Consider the graph; it passes the horizontal line test.

• f(x) = eˣ: The exponential function is strictly increasing across its entire domain, making it one-to-one. The graph clearly satisfies the horizontal line test.

Examples of Functions that are NOT One-to-One

It's equally important to understand how to identify functions that are not one-to-one.

• f(x) = x²: This function is not one-to-one because, for example, f(2) = f(-2) = 4. Two different inputs produce the same output. The graph fails the horizontal line test.

• f(x) = sin(x): The sine function is periodic; it repeats its values infinitely many times. Therefore, it's not one-to-one over its entire domain. Many horizontal lines intersect the sine wave multiple times.

• f(x) = x² - 4x + 4: This function can be factored as (x-2)². Notice that f(0) = f(4) = 4. Hence, it's not one-to-one.

• f(x) = |x|: The absolute value function is not one-to-one because, for example, |2| = |-2| = 2.

The Inverse Function and its Relationship to One-to-One Functions

Only one-to-one functions have inverse functions. The inverse function, denoted as f⁻¹(x), essentially "undoes" the action of the original function. If f(a) = b, then f⁻¹(b) = a. The domain of f becomes the codomain of f⁻¹, and vice versa. The graphs of f(x) and f⁻¹(x) are reflections of each other across the line y = x.

For example, the inverse of f(x) = x + 2 is f⁻¹(x) = x - 2. The inverse of f(x) = 2x is f⁻¹(x) = x/2. However, f(x) = x² does not have an inverse function over its entire domain because it's not one-to-one. However, if we restrict the domain to, say, non-negative real numbers (x ≥ 0), then it does have an inverse: f⁻¹(x) = √x.

Determining One-to-One Functions from Relations

When presented with a relation, whether as a set of ordered pairs, a table, or a graph, you can determine if it's a one-to-one function by checking if each output (y-value) is associated with exactly one input (x-value). If any output is associated with more than one input, the relation is not a one-to-one function.

Conclusion: Mastering One-to-One Functions

The concept of a one-to-one function is crucial in various mathematical and computational contexts. The ability to identify one-to-one functions is essential for understanding inverse functions, solving equations, and analyzing function behavior. By applying the horizontal line test, the algebraic approach, or by analyzing the function's properties, you can effectively determine whether a given function or relation satisfies the strict condition of one-to-one mapping. Remember, the key characteristic is that each output is uniquely linked to only one input. Understanding this ensures a strong foundation in your mathematical journey.