Problem 10 - College Algebra

Question

determine whether each relation is a function. Give the domain and range for each relation. $$ [(4,1),(5,1),(6,1)] $$

Step-by-Step Solution

Answer

Yes, the relation is a function. The domain is {4, 5, 6} and the range is {1}.

Step 1: Determine if the relation is a function

A relation is a function if every x-value (input) has exactly one y-value (output). Therefore, check if every x-value in the relation has only one associated y-value. In this case, the x-values are 4, 5, and 6. Each of these x-values has exactly one associated y-value, which is 1. Therefore, this relation is a function.

Step 2: Finding the domain

The domain is the set of all possible x-values (inputs). In this relation, the x-values are 4, 5, and 6. Therefore, the domain of this function is {4, 5, 6}.

Step 3: Finding the range

The range is the set of all possible y-values (outputs). In this relation, the y-value for all x-values is 1. Therefore, the range of this function is {1}.

Key Concepts

Function Definition

At its core, a function in algebra represents a special relationship where each input (often represented as x) is related to exactly one output (often represented as y). Imagine a function as a machine: for every item you put in, you get a specific item out. If you put the same item in, you'll always get the same item out. It's this consistent pair-up of inputs and outputs that makes a set of pairs a function.

For example, when you see an ordered pair like (4, 1), think of it as a simple instruction that says 'when the input is 4, the output is 1'. You can have many such instructions in a function, but crucially, you cannot have two different outputs for the same input. That would be like a vending machine giving you soda and chips when you only selected soda, and in math, that’s a no-go for functions.

Verifying a Function

To verify if a relation is a function, check each x-value to ensure it corresponds to one and only one y-value. If you find an x-value paired with multiple y-values, your relation is not a function. However, the reverse isn't true—multiple x-values can share the same y-value. That's like multiple vending machines containing the same type of soda—a perfectly acceptable scenario in the world of functions.

Domain and Range

The domain of a function is a fancy term for all the possible inputs (x-values) that a function can accept without breaking the mathematical 'rules’. In simpler terms, it's like a list of all the different coins and bills a vending machine accepts. If you've got an input that's not part of the function’s domain, the 'mathematical machine' cannot work with it, and it spits it back out.

Conversely, the range of a function is the set of all possible outputs (y-values) that come out of the function. Back to the vending machine analogy, it's like the assortment of snacks and drinks that can possibly tumble out when you punch in your selection. If an output is not part of the function’s range, it's like a vending machine that promises snacks it doesn't have—quite frustrating and not how a well-defined function should behave.

Identifying Domain and Range

When identifying the domain of a given list of ordered pairs, you look for all the unique x-values. In our example with the pairs \(\{(4,1),(5,1),(6,1)\}\), the domain is \(\{4, 5, 6\}\). The range is much easier in this case since each x-value is associated with the same y-value. Hence, the range is simply \(\{1\}\). It's worth noting that when listing elements of the domain or range, you use set notation, where each element is unique and order doesn’t matter.

X-Value and Y-Value

In algebra, ordered pairs are often used to represent the relationship between two things. Each pair has two parts: the x-value (the first number) and the y-value (the second number). You can think of the x-value as the input or the question asked in a problem, while the y-value is the output or the answer to that question.

The x-value holds quite a bit of responsibility. It determines what the y-value will be, following the rules of the function. In the ordered pair (4,1), the x-value is 4. It’s like asking a question: 'What do I get when I input 4 into the function?' The y-value, in this case 1, answers: 'You get 1!'

The Role of X and Y

Understanding the role of the x-value and y-value is crucial because it helps you picture what a function does. It shows you how changing the input (x) can change the output (y). This relationship is visualized graphically on a coordinate plane, where x represents the horizontal movement and y represents the vertical. It's this dance between x and y that gives us the curves and lines you see on graphs, representing functions in a visual and often easier to understand form.