Problem 3 - Algebra 2 and Trigonometry

Question

In \(3-5 :\) a. Explain why each set of ordered pairs is or is not a function. b. List the elements of the domain. c. List the clements of the range. $$ \\{(1,1),(2,4),(3,9),(4,16)\\} $$

Step-by-Step Solution

Answer

(a) It is a function. (b) Domain: \(\{1, 2, 3, 4\}\). (c) Range: \(\{1, 4, 9, 16\}\).

Step 1: Determine if the Relation is a Function

To determine if the set of ordered pairs represents a function, check if each input (first element of the ordered pair) is associated with exactly one output (second element of the ordered pair). In the given set \(\{(1,1),(2,4),(3,9),(4,16)\}\), each unique input (1, 2, 3, and 4) is paired with a unique output (1, 4, 9, and 16 respectively), meaning it satisfies the definition of a function.

Step 2: List the Domain

The domain of a function consists of all the first elements (inputs) from the ordered pairs. For the set \(\{(1,1),(2,4),(3,9),(4,16)\}\), the domain is the set \(\{1, 2, 3, 4\}\).

Step 3: List the Range

The range of the function consists of all the second elements (outputs) from the ordered pairs. For the set \(\{(1,1),(2,4),(3,9),(4,16)\}\), the range is \(\{1, 4, 9, 16\}\).

Key Concepts

Domain

In the realm of functions, the **domain** is a fundamental concept that refers to the complete set of possible input values you can use in a function. Think of it as all the potential ingredients you might consider when making a recipe. In mathematical terms, each member of the domain is referred to as an "input" or the "first element" in a series of ordered pairs.

For example, in the set of ordered pairs given each pair has a first element that represents the input. In this case, the numbers 1, 2, 3, and 4 are your domain because they're the values that are being plugged into the function. Essentially, the domain encompasses everything you put into a function to get a desired result.

Range

After understanding what you can put into a function, the **range** tells you what you can get out of it. The range consists of all possible outputs, or results, that you obtain from the function. Think of the range as all the possible meals you can create with your given ingredients.

In the same set of ordered pairs The numbers 1, 4, 9, and 16 represent the range. These numbers are the outputs of the function, and they are the result of each input from the domain being processed. Simply put, the range is where all the action ends, showing what comes out of your function.

Ordered Pairs

Understanding **ordered pairs** is crucial when dealing with functions. An ordered pair is typically written in parentheses like this: In its essence, an ordered pair links one member from the domain to a member in the range.

They're how we map inputs to outputs. Ordered pairs are usually composed of two components - the first element (the input) and the second element (the output). For the set Each pair shows a unique mapping: like 1 to 1, 2 to 4, and so on. This relation clearly defines how inputs from the domain are paired with outputs in the range. Thus, understanding ordered pairs helps us see how a function operates, linking each input to precisely one output.