Problem 7 - 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 ? $$ \begin{array}{ll} \text { Input } & \text { Output } \\ \hline 0 & 2 \\ 1 & 3 \\ 2 & 2 \\ 3 & 3 \\ 4 & 2 \end{array} $$

Step-by-Step Solution

Answer

The given relationship is a function. The domain of this function is \{0, 1, 2, 3, 4\} and the range of the function is \{2, 3\}. However, this is not a 1-to-1 function.

Step 1: Analyzing the function

To verify if the relationship provided is a function, check that every input corresponds to exactly one output. Observing the given Input-Output table, it is noted that every input is linked to just one output, thus this relationship is indeed a function.

Step 2: Finding the Domain

The domain is a set of all possible input values. Examining the 'Input' column in the table, the domain of this function will be the set of all input values which is \{0, 1, 2, 3, 4\}.

Step 3: Finding the Range

The range is a set of all possible output values. By checking 'Output' column in the table, the range of this function is \{2, 3\}.

Step 4: Checking for a 1-to-1 function

To determine if it is a 1-to-1 function, every element of the domain should correspond to exactly one unique element of the range. In the provided table, the inputs 0, 2, 4 all map to the same output 2 and the inputs 1, 3 map to the same output 3. Therefore, this is not a 1-to-1 function as multiple domain values map to the same range value.

Key Concepts

Input-Output Relationship

The concept of an input-output relationship is fundamental to understanding functions in calculus. Think of a function as a special type of machine in a factory: for every raw material (input) you put into the machine, you get a finished product (output). In mathematical terms, a function takes an input, applies a rule or formula to it, and then produces an output. It's crucial that for each input, there is one and only one output to qualify as a function. This exclusivity is what distinguishes functions from other types of relationships.

Our textbook exercise illustrates this relationship perfectly. It presents us with different inputs (0, 1, 2, 3, 4) and their respective outputs after being 'processed' by the function we're examining. In essence, to determine if this relationship is a function, we've verified that each individual input has a single, unique output, which aligns with our core definition of a function.

Domain and Range of a Function

Understanding the domain and range of a function is akin to knowing what ingredients can go into a recipe (domain) and what kinds of dishes can come out of it (range). The domain of a function consists of all the possible inputs that the function can accept without any issues, such as division by zero or taking the square root of a negative number in real numbers. On the flip side, the range is all the possible outputs that can result from plugging the domain values into our function.

In our exercise's context, we've determined that the domain is the set \( \{0, 1, 2, 3, 4\} \) based on the inputs available. Similarly, we've found the range to be \( \{2, 3\} \) by looking at the output values presented. Spotting the domain and range helps us understand the breadth and scope of the function's action.

1-to-1 Function

A 1-to-1 function, or injective function, is a bit like having VIP passes for a concert; each pass will let in exactly one person, no duplicates allowed. In mathematical terms, a function is 1-to-1 if each input produces a unique output, and no two different inputs produce the same output. To visualize this, imagine that every person (input) has their own unique seat (output) in a theater. If two people are assigned to one seat, it's not a 1-to-1 function.

The textbook example we're reviewing shows us that the function is not 1-to-1. Why? Because different inputs, specifically 0, 2, and 4, all produce the same output, 2. This violates our 'one person, one seat' rule, so it's clear that multiple inputs are 'sharing' an output, disqualifying it as a 1-to-1 function.

Function Analysis

Function analysis is essentially detective work where we examine different aspects of a function to understand its behavior thoroughly. It involves looking at the input-output relationship for functionality, identifying the set of possible inputs (domain), possible outputs (range), and checking for unique relationships such as whether the function is 1-to-1.

In our step-by-step solution, we conducted a basic function analysis: we established that the presented relationship is indeed a function, determined its domain and range, and assessed its 1-to-1 status. Why do we do this? Function analysis helps us anticipate what a function will do with certain inputs, understand its limitations, and its inherent characteristics, such as whether it’s invertible, which is directly related to the function being 1-to-1.