Problem 19 - Algebra 1

Question

Decide whether the relation is a function. If it is a function, give the domain and the range. $$ \begin{array}{|c|c|}\hline \text { Input } & \text { Output } \\\\\hline 1 & 1 \\\\\hline 3 & 2 \\\\\hline 5 & 3 \\\\\hline 7 & 1 \\\\\hline\end{array} $$

Step-by-Step Solution

Answer

The given relation is a function. Its domain is \{1, 3, 5, 7\} and its range is \{1, 2, 3\}.

Step 1: Identify if the Relation is a Function

Observe that each input in the given relation corresponds to exactly one output, which implies that the given relation is indeed a function.

Step 2: Identify the Domain of the Function

The domain of a function consists of all possible input values. Looking at the 'Input' column in the relation table, the domain of the function is \{1, 3, 5, 7\}.

Step 3: Identify the Range of the Function

The range of a function consists of all possible output values. Looking at the 'Output' column in the relation table, the range of the function is \{1, 2, 3\}.

Key Concepts

Domain and Range

When exploring functions, two fundamental concepts to grasp are the domain and range. These terms help us understand which values a function can accept and return. The **domain** refers to all possible input values for the function. In simpler terms, it answers the question: "What can I put into this function?" For example, if we consider a function where the inputs are 1, 3, 5, and 7, then these values make up the domain.
On the other hand, the **range** pertains to all possible outputs produced by the function. It's what the function can "output" after processing the input values. For instance, if these input values produce outputs of 1, 2, and 3, then these are considered the range of the function.
The domain and range give us a full picture of how a function behaves and what it can do. They tell us both the extent and limitations of the function's capability.

Relations

A relation is a set of ordered pairs, like (1, 1), (3, 2), etc. In mathematics, quite a few scenarios can be depicted as relations—connections between two sets of information. For instance, the relation between students and their heights, or days of the week and temperatures.
Now, when we talk about a **function**, it’s a special kind of relation. Specifically, each input in the function relation must connect to exactly one output. Think of it as a vending machine: for one coin inserted, you get exactly one drink, not multiple. If a relation has this one-to-one character where each input is paired uniquely with an output, then it qualifies as a function.

• In our example, each input (1, 3, 5, 7) leads to a unique output, which confirms it is a function.
• This unique assignment makes functions predictable and manageable, unlike general relations.

Input-Output Table

An Input-Output table is a handy tool to visually organize the relationship between inputs and outputs of a function or a relation. It consists of two columns, where one column represents the inputs and the other represents the outputs.
Using an **input-output table**, we can easily see how each specific input is connected to an output. This method of organization helps simplify the identification of domains and ranges, and makes it easier to figure out if the relation is a function.
For example, in the table given:

• Inputs: 1, 3, 5, 7
• Outputs: 1, 2, 3

It's visibly clear how each input is linked to an output. Through this "mapping" method, the table quickly shows us if there are any overlapping outputs per input, a crucial step in verifying whether a relation is a function.