Select All Of The Following Graphs Which Are One-to-one Functions.

Select All of the Following Graphs Which Are One-to-One Functions

Understanding one-to-one functions is crucial in various mathematical fields, including calculus, linear algebra, and beyond. This comprehensive guide will delve deep into the concept of one-to-one functions, exploring their properties, how to identify them graphically and algebraically, and providing practical examples to solidify your understanding. We'll also explore the inverse function, a concept inextricably linked to one-to-one functions.

What is a One-to-One Function?

A one-to-one function, also known as an injective function, is a function where each element in the range of the function corresponds to exactly one element in the domain. In simpler terms, no two different inputs (elements in the domain) produce the same output (element in the range). This contrasts with many-to-one functions where multiple inputs can map to the same output.

Key Characteristics of One-to-One Functions:

• Uniqueness of Output: Every output value has only one corresponding input value.
• Horizontal Line Test: A graph represents a one-to-one function if and only if no horizontal line intersects the graph more than once. This is a powerful visual tool for identifying one-to-one functions.
• Injective Mapping: Mathematically, a function f: A → B is one-to-one if and only if for all x₁ and x₂ in A, if f(x₁) = f(x₂), then x₁ = x₂. This means that if two inputs produce the same output, then those inputs must be identical.

Identifying One-to-One Functions Graphically

The horizontal line test is the most straightforward method for determining whether a function is one-to-one from its graph. If any horizontal line intersects the graph at more than one point, then the function is not one-to-one. Conversely, if every horizontal line intersects the graph at most once, the function is one-to-one.

Let's consider some examples:

Example 1: A One-to-One Function

Imagine a graph that represents a strictly increasing linear function, such as f(x) = x + 2. No horizontal line can intersect this graph more than once. Therefore, f(x) = x + 2 is a one-to-one function.

Example 2: A Many-to-One Function

Consider the graph of a parabola, such as f(x) = x². A horizontal line drawn above the x-axis will intersect the parabola at two points. This indicates that f(x) = x² is not a one-to-one function because multiple x values map to the same y value (e.g., f(2) = f(-2) = 4).

Example 3: A More Complex Scenario

Consider a more complex function whose graph has both increasing and decreasing sections. Carefully examining the graph using the horizontal line test is crucial. If any horizontal line intersects the graph more than once, the function is not one-to-one.

Identifying One-to-One Functions Algebraically

While the graphical method is intuitive, algebraic methods are necessary when dealing with functions whose graphs are difficult or impossible to visualize. The algebraic approach involves employing the definition of a one-to-one function directly: assume f(x₁) = f(x₂), and then show that this implies x₁ = x₂.

Example 4: Algebraic Proof

Let's prove that f(x) = 3x + 5 is a one-to-one function algebraically.

1. Assume: f(x₁) = f(x₂)
2. Substitute: 3x₁ + 5 = 3x₂ + 5
3. Simplify: 3x₁ = 3x₂
4. Solve for x: x₁ = x₂

Since we've shown that f(x₁) = f(x₂) implies x₁ = x₂, the function f(x) = 3x + 5 is one-to-one.

Example 5: A Function that is NOT One-to-One (Algebraic Proof)

Let's show that f(x) = x² - 4 is not a one-to-one function algebraically.

1. Assume: f(x₁) = f(x₂)
2. Substitute: x₁² - 4 = x₂² - 4
3. Simplify: x₁² = x₂²
4. Solve for x: x₁ = ±x₂

This shows that x₁ can be equal to x₂ or -x₂, meaning that different inputs can produce the same output. Therefore, f(x) = x² - 4 is not one-to-one.

The Inverse Function and One-to-One Functions

A crucial aspect of one-to-one functions is their connection to inverse functions. Only one-to-one functions have inverse functions. The inverse function, denoted as f⁻¹(x), essentially "undoes" the operation of the original function. If f(a) = b, then f⁻¹(b) = a.

Finding the Inverse Function:

To find the inverse function of a one-to-one function, follow these steps:

1. Replace f(x) with y: This makes the equation easier to manipulate.
2. Swap x and y: This reflects the inverse relationship.
3. Solve for y: This isolates y in terms of x.
4. Replace y with f⁻¹(x): This formally defines the inverse function.

Example 6: Finding an Inverse Function

Let's find the inverse function of f(x) = 2x - 1.

1. y = 2x - 1
2. x = 2y - 1
3. x + 1 = 2y
4. y = (x + 1)/2
5. f⁻¹(x) = (x + 1)/2

Therefore, the inverse function of f(x) = 2x - 1 is f⁻¹(x) = (x + 1)/2.

Applications of One-to-One Functions

One-to-one functions have numerous applications across various fields:

• Cryptography: One-to-one functions are essential in encryption algorithms, ensuring that each plaintext message maps to a unique ciphertext.
• Coding Theory: In data compression and error correction codes, one-to-one mappings are used to represent data efficiently and reliably.
• Computer Science: Injective functions are crucial in data structures and algorithms, guaranteeing unique representations of data elements.
• Calculus: The concept of one-to-one functions is fundamental for understanding derivatives and integrals, especially in relation to inverse functions.

Advanced Concepts and Further Exploration

This exploration of one-to-one functions provides a solid foundation. For a deeper understanding, you can explore:

• Strictly Monotonic Functions: Strictly increasing or decreasing functions are always one-to-one.
• Bijective Functions: A function that is both one-to-one (injective) and onto (surjective) is called bijective. Bijections are crucial in establishing a one-to-one correspondence between sets.
• Cardinality of Sets: One-to-one correspondences are used to compare the sizes of infinite sets.

By mastering the concepts presented here, you'll be well-equipped to tackle more advanced mathematical topics and applications involving functions and their properties. Remember to practice identifying one-to-one functions graphically and algebraically – the more you practice, the better your understanding will become. The horizontal line test, in particular, is an invaluable visual tool that will aid your intuition. Don't hesitate to explore additional resources and examples to further solidify your grasp of this fundamental mathematical concept. Remember to always check your work carefully and consider the implications of a function being, or not being, one-to-one within the context of the problem you are solving.