Problem 18 - Algebra 1: Concepts and Skills

Question

Determine 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

Yes, the relation is a function. The domain of the function is {1,3,5,7} and the range of the function is {1,2,3}.

Step 1: Verify if Relation is a Function

Look at the input values in the relation. In this case, the input values are {1,3,5,7}. If each input corresponds to exactly one output then the relation is a function. The input 1 corresponds to output 1, input 3 corresponds to output 2, input 5 corresponds to output 3, and finally, input 7 corresponds to output 1. Hence, we see that each input corresponds to exactly one output. Therefore, the relation is a function.

Step 2: Identify the Domain

The domain of a function constitutes all the possible input values. Looking at the relation, we see that the possible input values are {1,3,5,7}. Therefore, this is the domain of the function.

Step 3: Identify the Range

The range consists of all the possible output values of a function. In this case, the output values are {1,2,3}. So, this is the range of the function.

Key Concepts

Domain and Range

Understanding the concepts of domain and range is essential when exploring functions in mathematics. Imagine you have a machine that takes in certain numbers and spits out others. The domain is like the list of ingredients you can put into the machine – specifically, it's all the potential input values that can be used in a function. For the function given in the exercise, we identified the domain as {1, 3, 5, 7} since these are all the inputs that have an associated output.

The range, on the other hand, is like the products that come out of the machine: it includes all the possible outputs that the function can produce. From our example, the outputs are 1, 2, and 3. Hence, the range of this function is {1, 2, 3}. It's important to note that while different inputs may yield the same output, the range includes each output value just once, excluding any repetition.

Function Definition

A function is a special type of relation where each input is related to exactly one output. Imagine a strict one-to-one rule where every participant (input) brings a unique dish (output) to a potluck dinner; no duplicates are allowed in terms of combinations. We can represent functions in many ways, such as equations, graphs, or, as in our exercise, a table of input-output pairs.

In the given table from our exercise, for a relation to be a function, each input must correspond to only one output. The table meets this criterion since each input number uniquely matches up with a single output number. This is why, upon examining the input-output pairs {1, 1}, {3, 2}, {5, 3}, and {7, 1}, we determine that this relation indeed qualifies as a function. This definition helps ensure a consistent association between inputs and outputs, which is fundamental for functions.

Input-Output Relationship

Exploring the input-output relationship involves understanding how each entry (input) in a function is mapped to a corresponding exit (output). A good analogy is to think of this relationship as a dance partnership where each lead (input) is paired with precisely one follow (output), creating a distinct dance couple.

In the context of the exercise, our function illustrates this principle: each input has one and only one designated output. An easy check for whether a relation is a function is to look for duplicate input values with different outputs; if you find any, you've stepped onto the territory of a non-function. Since our example avoids this pitfall, with the inputs being {1, 3, 5, 7} and each having a unique output, we can say with confidence that we have a functional input-output relationship.