Problem 3 - College Algebra Essentials

Question

Determine whether each relation is a function. Give the domain and range for each relation. $$ \\{(3,4),(3,5),(4,4),(4,5)\\} $$

Step-by-Step Solution

Answer

The given relation is not a function. The domain of this relation is \{3,4\}, and the range is \{4,5\}.

Step 1: Determining if the relation is a function

Pick each pair in the relation: (3,4), (3,5), (4,4) and (4,5). If for each x there is only one unique y, then it is a function. If not, it isn't. Here, 3 corresponds to 4 and 5, and similarly 4 also relates to 5 and 4. Hence this relation is not a function because the same input corresponds to different outputs.

Step 2: Determining the domain

The domain of a function or relation is the set of all x-coordinates or input of the ordered pairs. In this case, the domain includes 3 and 4.

Step 3: Determining the range

The range of a function or relation is the set of all y-coordinates or output of the ordered pairs. In this case, the range includes 4 and 5.

Key Concepts

Determining Functions

Understanding whether a relation is a function is crucial in the world of algebra. A function is defined by a special relationship between two sets: each element of the first set, often referred to as the input or domain, is associated with exactly one element of the second set, known as the output or range.

When looking at a set of ordered pairs, like \(\{(3,4),(3,5),(4,4),(4,5)\}\), the crux lies in identifying the instances where an input, or the first number in each pair, corresponds to more than one output. In this case, the number 3 corresponds to both 4 and 5, and 4 corresponds to 4 and 5 as well. This repetition indicates that the relation isn't a function, because a true function would not have an input point to different outputs. Remember, for a relation to be a function, each input value must match with exactly one output value.

Domain and Range

The domain and range of a relation or function are foundational concepts in algebra that describe the possible values of inputs and outputs. The domain consists of all the first elements in the ordered pairs—these are your possible inputs. In our set of ordered pairs, the domain is composed of the values 3 and 4, as they are the first numbers in each pair.As for the range, it encompasses all the second elements of the ordered pairs, which represent the possible outputs. Here, the range is made up of the numbers 4 and 5. Understanding domain and range is essential, as these sets provide a view of what values are in play within a relation or function. When describing the domain and range, make sure to list the unique values; this ensures clarity and helps in visualizing the relation or function.

Ordered Pairs

Ordered pairs are a fundamental component of functions and relations in algebra. They consist of two elements written in a specific order, usually within parentheses, such as \( (x, y) \). The first element, x, represents the input or the domain element, while the second, y, denotes the output or the range element. The order matters because the first element of the pair is understood to map to the second element, establishing the relationship between the two sets.In examining a set of ordered pairs, such as \(\{(3,4),(3,5),(4,4),(4,5)\}\), it's important to look for patterns or unique correspondences between the x and y values to determine the structure and characteristics of the relation. The notation of ordered pairs is a clear and concise way to express these relationships and is a key stepping stone in grasping more complex concepts in algebra.