Problem 17 - Algebra 1

Question

Decide whether the relation is a function. If it is a function, give the domain and the range. $$ \begin{array}{|c|c|}\hline \text { Input } & \text { Output } \\\\\hline 0 & 2 \\\\\hline 1 & 4 \\\\\hline 2 & 6 \\\\\hline 3 & 8 \\\\\hline\end{array} $$

Step-by-Step Solution

Answer

The given relation is a function. The domain is {0, 1, 2, 3} and the range is {2, 4, 6, 8}.

Step 1: Identify Whether the Relation is a Function

Look at each input in the relation. If any input corresponds to more than one output, then it's not a function. In this case, each input (0, 1, 2, 3) is mapped to exactly one output (2, 4, 6, 8), so the relation is a function.

Step 2: Identify the Domain

The domain of a function is the set of all possible input values. Looking at the relation, the inputs are clearly {0, 1, 2, 3}. Therefore, the domain is {0, 1, 2, 3}.

Step 3: Identify the Range

The range of a function is the set of all possible output values. Looking at the relation, the outputs are {2, 4, 6, 8}. So, the range is {2, 4, 6, 8}.

Key Concepts

Understanding the Domain of a Function

The domain of a function is a fundamental concept in math, crucial to understanding how inputs relate to outputs. When we talk about the domain, we are discussing the complete set of all possible input values, which are also known as the independent variable values. For example, when provided with a list of ordered pairs, as in the given exercise with inputs

• 0
• 1
• 2
• 3

we can identify the domain by simply listing these input values since they represent every possible input that the function can accept. It is like asking, "What values can I plug into the function?"
In equations, sometimes we might have restrictions due to the nature of operations like square roots or division. But in this case, with clear numerical pairs, the domain is straightforward and encompasses exactly those four values.

Exploring the Range of a Function

The range of a function complements the concept of the domain. While the domain addresses possible inputs, the range focuses on possible outputs, or dependent variable values. Simply put, the range is the set of values that the function can produce.
In the exercise example, the outputs are namely:

• 2
• 4
• 6
• 8

indicating that these are the only values the function associates with its domain inputs. To derive the range from a table or a list of pairs, we scan the output side to compile a complete list of unique results.
Thus, the range effectively answers the question, "What values might I get out of the function." It helps us understand the behavior and limits of the function's outputs depending on various inputs.

Defining Relation as Functions

In mathematics, a relation pairs inputs with outputs. However, to be considered a function, each input must correspond to exactly one output. This restriction is what differentiates a general relation from a function.
In the exercise step, each input (0, 1, 2, 3) mapped to a unique output (2, 4, 6, 8), validating it as a function. It’s crucial to recognize this because if any input were tied to multiple outputs, the relation would not be a function.
This single association characteristic is essential since functions need to be predictable—each input producing one specific output every time it is used.

• Functions help in creating mathematical models.
• They allow for effective computation and prediction.
• Understanding relations vs. functions forms the basis of many algebraic processes.

Grasping the difference between a mere relation and a function ensures you accurately interpret various mathematical scenarios.