Problem 8 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

For Problems 7 through 9 determine whether the relationship described is a function. If the relationship is a function, (a) what is the domain? the range? (b) is the function 1 -to- 1 ? $$ \begin{array}{ll} \text { Input } & \text { Output } \\ \hline \sqrt{2} & 2 \\ \sqrt{3} & 3 \\ \sqrt{5} & 5 \\ \sqrt{6} & 6 \end{array} $$

Step-by-Step Solution

Answer

The relationship described is indeed a function. The domain of this function is \(\sqrt{2}\), \(\sqrt{3}\), \(\sqrt{5}\), \(\sqrt{6}\), and the range is 2, 3, 5, 6. The function is a 1-to-1.

Step 1: Understand the definition of a function

A relationship between two quantities is defined as a function if every input has exactly one output. Looking at the problem, we see that every input (\(\sqrt{2}\), \(\sqrt{3}\), \(\sqrt{5}\), \(\sqrt{6}\)) has exactly one corresponding output (2, 3, 5, 6), so it is a function.

Step 2: Identify the domain of the function

The domain of a function is the set of all possible input values. In this case the domain is the set of all inputs, so the domain is \(\sqrt{2}\), \(\sqrt{3}\), \(\sqrt{5}\), \(\sqrt{6}\).

Step 3: Identify the range of the function

The range of a function is the set of all possible output values. Looking at the problem, we see that the range is the set of all outputs, so the range is 2, 3, 5, 6.

Step 4: Determine if the function is 1-to-1

A function is described as 1-to-1, or injective, if every output is generated by exactly one input. Here, no two inputs map to the same output, so the function is 1-to-1.

Key Concepts

Domain

In mathematics, when discussing functions, the **domain** is an essential concept. The domain refers to the complete set of possible input values that a function can accept. Understanding the domain is crucial because it outlines the "inputs" for the function. Features of the domain include:

• The domain consists solely of inputs that will not lead to undefined situations in the function.
• Typically represented using set notation or listing actual values when possible.
• Sometimes restrictions on domains occur due to mathematical operations like division by zero or taking square roots of negative numbers.

In our problem, the domain comprises the specific values:

• \( \sqrt{2} \)
• \( \sqrt{3} \)
• \( \sqrt{5} \)
• \( \sqrt{6} \)

Each of these inputs is a unique non-negative square root of a prime or composite number, suitable for the function described in the problem.

Range

While the domain is all about inputs, the **range** relates to the outputs of a function. In simpler terms, it’s the set of all possible results that a function can produce from its domain. Here are some essential points:

• Determining the range requires observing the outcomes derived from each input within the domain.
• It often requires solving the function for each possible input or understanding the logic or pattern of the outputs.
• Writing the range usually involves listing all possible outcome values that the function might yield.

In the given situation, the range of this function consists of the outputs:

• 2
• 3
• 5
• 6

Each of these numbers corresponds directly to one of our domain values. Since each input matches a unique output, determining the range is straightforward in this scenario.

Injective Function

An **injective function**, also known as a one-to-one function, is a type of function where each output value is uniquely tied to one input. Injective functions are crucial because they guarantee that information is not lost when transitioning from input values to output values. Here’s what you need to know:

• Each element in the range is mapped by exactly one element in the domain.
• If different inputs lead to the same output, the function is not injective.
• Injective functions simplify reversing the mapping process since each output corresponds to a unique input.

In the given example, every output from the function (\(2, 3, 5, 6\)) is determined by a unique input (\(\sqrt{2}, \sqrt{3}, \sqrt{5}, \sqrt{6}\)). Consequently, the function can be described as injective or one-to-one. This property ensures that each input value corresponds to a distinct output value without any overlap or repetition.