Problem 6 - College Algebra

Question

determine whether each relation is a function. Give the domain and range for each relation. $$ [(10,4),(-2,4),(-1,1),(5,6)] $$

Step-by-Step Solution

Answer

The given relation is a function. Its domain is {10,-2,-1,5} and its range is {4,1,6}.

Step 1: Check if Relation is a Function

To check if the relation given by the set of points {(10,4),(-2,4),(-1,1),(5,6)} is a function, we need to check if each 'x' value (first element of the pair) is paired with exactly one 'y' value (second element of the pair). In this set, every 'x' value only appears once, which means it's paired with exactly one 'y' value. So, this relation is indeed a function.

Step 2: Identify Domain

The domain of a function is the set of all possible 'x' values. For this function, the domain (set of 'x' values) would be {10,-2,-1,5}, listed in order of the pairs given.

Step 3: Identify Range

The range of a function is the set of all possible 'y' values. For this function, the range (set of 'y' values) would be {4,1,6}, listed in order of the pairs given.

Key Concepts

Relations and Functions

In mathematics, a **relation** is a connection between sets of numbers, typically expressed as pairs. In the given exercise, the relation is presented by a set of ordered pairs: \((10,4), (-2,4), (-1,1), (5,6)\). Each pair links an 'x' value, which is the first element, with a 'y' value, which is the second element. For a relation to be classified as a **function**, each 'x' value in these pairs must be linked to exactly one 'y' value. This implies that no 'x' value repeats with a different 'y' value. In other words:- Each 'x' has one and only one 'y' associated with it.- In this particular relation, every 'x' (10, -2, -1, 5) is unique, affirming that it meets the criteria of a function.

Understanding the Domain

The **domain** of a function is a collection of all input values ('x' values) that a function can accept. From our list of pairs, the domain is derived by compiling all the different 'x' values present in the relation. To determine the domain, look at each pair and note the 'x' values:

• (10, 4)
• (-2, 4)
• (-1, 1)
• (5, 6)

These 'x' values form the domain \({10, -2, -1, 5}\). It's crucial always to list them regardless of how they are ordered in the pairs, as each represents a valid input for the function.

Exploring the Range

The **range** is comprised of all possible output values ('y' values) that a function can produce from its inputs. For the given set of pairs, the range reflects the diverse 'y' values extracted from each part of the relation. To find the range, observe each pair for its 'y' value:

• (10, 4)
• (-2, 4)
• (-1, 1)
• (5, 6)

The 'y' values \(4, 1, \) and \(6\) make up the range \({4, 1, 6}\). Notice that the value \(4\) occurs twice; however, it is only listed once in the range since we focus solely on distinct values regardless of their repetition in the pairs.