Which Of The Following Graphs Represents A One-to-one Function

Which of the Following Graphs Represents a One-to-One Function?

Understanding functions is fundamental in mathematics, and within the realm of functions, one-to-one (or injective) functions hold a special place. This article will delve deep into the concept of one-to-one functions, explaining what they are, how to identify them graphically, and how to distinguish them from other types of functions. We'll explore various examples and provide a robust framework for determining whether a given graph represents a one-to-one function.

What is a One-to-One Function?

A function, in its simplest form, is a relationship where each input (x-value) corresponds to exactly one output (y-value). However, a one-to-one function, also known as an injective function, imposes an additional constraint: each output (y-value) must also correspond to exactly one input (x-value). In other words, no two different inputs can produce the same output.

Think of it like a machine: a regular function takes an input and produces an output. A one-to-one function is a special kind of machine where no two different inputs ever produce the same output. If you know the output, you can uniquely determine the input.

Graphical Identification of One-to-One Functions: The Horizontal Line Test

The easiest way to determine if a function is one-to-one from its graph is by applying the horizontal line test. This test is a simple visual check:

If any horizontal line intersects the graph at more than one point, the function is NOT one-to-one. Conversely, if every horizontal line intersects the graph at most once (meaning it intersects at only one point or not at all), then the function is one-to-one.

Let's illustrate this with some examples:

Example 1: A One-to-One Function

Consider the graph of the function f(x) = x³. If we draw several horizontal lines across the graph, we'll observe that each line intersects the curve at only one point. Therefore, this function is one-to-one. Every unique y-value corresponds to a unique x-value.

Visual Representation:

Imagine a smoothly increasing cubic curve. No horizontal line will ever intersect it more than once.

Example 2: A Function That is NOT One-to-One

Now, let's consider the graph of the parabola f(x) = x². This is a familiar U-shaped curve. If we draw a horizontal line above the x-axis (e.g., y = 4), we'll see that it intersects the parabola at two points. This means there are two distinct x-values (x = 2 and x = -2) that produce the same y-value (y = 4). Therefore, f(x) = x² is not a one-to-one function.

Visual Representation:

The parabola's symmetry is key here. A horizontal line intersecting the parabola above the vertex will always cross the parabola at two points, failing the horizontal line test.

Example 3: A Piecewise Function

Piecewise functions can be more challenging. Let's examine a piecewise function defined as:

f(x) = x, if x ≥ 0 -x, if x < 0

This function represents the absolute value function but with a different definition for positive and negative x values. Let's analyze it graphically:

The graph will consist of two rays: one with a positive slope for x ≥ 0 and one with a negative slope for x < 0, meeting at the origin (0,0). Any horizontal line above the x-axis will intersect the graph at two points, failing the horizontal line test. Hence, this function is not one-to-one.

Visual Representation:

The graph resembles a V-shape. Horizontal lines will intersect the V-shape twice for y > 0.

Example 4: A More Complex Function

Consider a function defined by the equation: f(x) = (x-1)(x-2)(x-3) + 1. This cubic polynomial can be tricky to visualize directly. To apply the horizontal line test, we'd need to carefully analyze its graph. A proper graph sketch or software tools will help determine the intersections of horizontal lines and subsequently decide if the function is one-to-one. In this case, the horizontal line test reveals it is not one-to-one because there exist horizontal lines that intersect the graph at more than one point.

Visual Representation:

This cubic function exhibits local maxima and minima. These are the points which allow horizontal lines to intersect it at more than one point.

The Importance of One-to-One Functions

One-to-one functions are crucial in several areas of mathematics:

• Inverse Functions: Only one-to-one functions have inverse functions. The inverse function essentially "undoes" the original function. If a function is not one-to-one, it cannot have a true inverse function.

• Cryptography: One-to-one functions are fundamental in cryptographic systems. The ability to uniquely determine the input from the output is vital for security.

• Calculus: In calculus, understanding one-to-one functions is important for topics like derivatives and integrals.

• Linear Algebra: Linear transformations are one-to-one if their matrices have full rank.

Distinguishing One-to-One from Other Types of Functions

It's important to distinguish one-to-one functions from other types of functions:

• Onto (Surjective) Functions: An onto function is one where every element in the codomain (the set of possible outputs) is mapped to by at least one element in the domain (the set of inputs). A function can be both one-to-one and onto (a bijection), but it doesn't have to be.

• Many-to-One Functions: These are functions where multiple inputs can map to the same output (as seen in Example 2). One-to-one functions are the opposite of many-to-one functions.

Advanced Techniques for Determining One-to-One Functions

While the horizontal line test is the most intuitive method for graphical identification, other techniques can be employed:

• Algebraic Approach: For functions defined algebraically, we can use algebraic manipulations to show that if f(x₁) = f(x₂), then x₁ = x₂. This directly demonstrates the one-to-one property.

• Derivative Test (for differentiable functions): If a function is strictly increasing or strictly decreasing over its entire domain, then it is one-to-one. This can be verified by examining the derivative of the function. If the derivative is always positive (strictly increasing) or always negative (strictly decreasing), the function is one-to-one. However, this test is only sufficient, not necessary. A one-to-one function doesn't need to be strictly monotonic.

• Analyzing the function's behavior: Understanding the function's properties, like symmetry, asymptotes, and intervals of increase/decrease, provides insightful information about whether the function is one-to-one.

Conclusion

Determining whether a graph represents a one-to-one function is a crucial skill in mathematics. The horizontal line test provides a simple yet effective visual method for this identification. Understanding the concept of one-to-one functions and their significance extends beyond basic function analysis, having important implications in various mathematical fields and applications. Mastering this concept solidifies your understanding of function properties and paves the way for tackling more complex mathematical problems. Remember to utilize the various techniques described above to analyze functions effectively and correctly determine their one-to-one nature.