Which Functions Are Invertible Select Each Correct Answer

Which Functions Are Invertible? Selecting the Correct Answers

Determining which functions are invertible is a crucial concept in mathematics, particularly in algebra and calculus. Understanding invertibility allows us to solve equations, understand transformations, and build more complex mathematical structures. This article will delve deep into the concept of invertible functions, providing clear explanations, examples, and strategies for identifying them. We will explore various function types and their characteristics to help you accurately select the correct answers when faced with questions about invertible functions.

What is an Invertible Function?

A function is invertible if and only if it's a one-to-one (injective) and onto (surjective) mapping. Let's break down these terms:

• One-to-one (Injective): A function is one-to-one if each element in the codomain (output) is mapped to by at most one element in the domain (input). In simpler terms, no two different inputs produce the same output. Formally, for all x₁ and x₂ in the domain, if f(x₁) = f(x₂), then x₁ = x₂. A common way to visualize this is the horizontal line test: if any horizontal line intersects the graph of the function at more than one point, the function is not one-to-one.

• Onto (Surjective): A function is onto if every element in the codomain is mapped to by at least one element in the domain. In other words, the range of the function is equal to its codomain. Every possible output value is achieved by at least one input value.

Only functions that are both one-to-one and onto are invertible. If a function lacks either property, it's not invertible. The inverse function, denoted as f⁻¹(x), reverses the mapping of the original function. If f(a) = b, then f⁻¹(b) = a.

Identifying Invertible Functions: A Step-by-Step Guide

To determine if a function is invertible, follow these steps:

1. Analyze the Function's Definition: Examine the rule or equation defining the function. Look for any clues that suggest a one-to-one or onto mapping. For instance, strictly increasing or decreasing functions are usually one-to-one.

2. Apply the Horizontal Line Test (for graphical representation): If you have a graph of the function, draw horizontal lines across the graph. If any horizontal line intersects the graph at more than one point, the function is not one-to-one, and therefore not invertible.

3. Check for One-to-One Property Algebraically: To prove a function is one-to-one algebraically, assume f(x₁) = f(x₂) and show that this implies x₁ = x₂.

4. Determine the Range (to check for Onto Property): Find the range of the function. If the range is equal to the codomain, the function is onto. If the range is a proper subset of the codomain, the function is not onto, and thus not invertible.

Examples of Invertible and Non-Invertible Functions

Let's illustrate these concepts with examples:

1. f(x) = 2x + 1 (Invertible)

• One-to-one: Assume f(x₁) = f(x₂). Then 2x₁ + 1 = 2x₂ + 1. Subtracting 1 from both sides and dividing by 2 gives x₁ = x₂. Therefore, the function is one-to-one.

• Onto: The function is a linear function with a non-zero slope. Its range is all real numbers, which is the same as its codomain (assuming the codomain is also all real numbers). Thus, the function is onto.

• Conclusion: Since f(x) = 2x + 1 is both one-to-one and onto, it is invertible. Its inverse is f⁻¹(x) = (x - 1)/2.

2. f(x) = x² (Not Invertible)

• One-to-one: This function fails the horizontal line test. For example, f(2) = 4 and f(-2) = 4. Since two different inputs (2 and -2) produce the same output (4), the function is not one-to-one.

• Onto: If the codomain is all real numbers, the function is not onto because it only produces non-negative outputs. The range is [0, ∞).

• Conclusion: Because f(x) = x² is neither one-to-one nor onto (when the codomain is all real numbers), it is not invertible. However, if we restrict the domain to x ≥ 0, then it becomes both one-to-one and onto (with codomain [0, ∞)), and its inverse would be f⁻¹(x) = √x.

3. f(x) = sin(x) (Not Invertible)

• One-to-one: The sine function is periodic, meaning it repeats its values infinitely. It fails the horizontal line test, therefore it's not one-to-one.

• Onto: The range of sin(x) is [-1, 1]. If the codomain is [-1, 1], then it is onto. However, if the codomain is all real numbers, it's not onto.

• Conclusion: Because sin(x) is not one-to-one, it is not invertible over its entire domain. However, by restricting the domain to [-π/2, π/2], we obtain a one-to-one and onto function (with codomain [-1, 1]), whose inverse is arcsin(x).

4. f(x) = eˣ (Invertible)

• One-to-one: The exponential function is strictly increasing; therefore, it passes the horizontal line test and is one-to-one.

• Onto: The range of eˣ is (0, ∞). If the codomain is (0, ∞), then the function is onto.

• Conclusion: The exponential function eˣ is both one-to-one and onto (when the codomain is (0, ∞)), making it invertible. Its inverse is the natural logarithm function, ln(x).

Advanced Considerations:

• Bijective Functions: A function that is both injective (one-to-one) and surjective (onto) is called bijective. Bijective functions are always invertible.

• Inverse Functions and Composition: If f(x) is invertible, then f(f⁻¹(x)) = x and f⁻¹(f(x)) = x. This means applying the function and its inverse in either order results in the original input.

• Applications of Inverse Functions: Inverse functions have numerous applications in various fields, including cryptography, computer graphics, and solving equations.

In Summary:

Determining which functions are invertible requires a clear understanding of one-to-one and onto mappings. By systematically analyzing the function's definition, applying the horizontal line test (graphically), and using algebraic methods to verify the one-to-one property and finding the range to check the onto property, you can accurately identify invertible functions and their inverses. Remember that restricting the domain of a function can sometimes make a non-invertible function invertible. Mastering this concept is essential for progressing in various mathematical disciplines and their applications. Practice with a variety of examples will solidify your understanding and improve your ability to correctly select the invertible functions from a given set.