Problem 8 - College Algebra

Question

determine whether each relation is a function. Give the domain and range for each relation. $$ |(-7,-7),(-5,-5),(-3,-3),(0,0)\\} $$

Step-by-Step Solution

Answer

The relation is a function. The domain of the function is \{-7, -5, -3, 0\} and the range is \{-7, -5, -3, 0\}

Step 1: Identify if the relation is a function

To check if a relation is a function, we need to ensure that for each individual x-value (or input), there is only one corresponding y-value (or output). In our relation, all x-values are distinct, meaning each x-value corresponds to a unique y-value. Thus, the relation is a function.

Step 2: Determine the domain of the function

The domain of a function is the complete set of possible x-values. By inspecting the relation or function, we can see that the domain is \(-7, -5, -3, 0\).

Step 3: Determine the range of the function

The range of a function is the complete set of possible y-values. By observing the relation or function, we can clearly see that the range is \(-7, -5, -3, 0\)

Key Concepts

Function Definition

Understanding the concept of a function is crucial in the study of algebra. Simply put, a function is a special kind of relation between two sets, typically referred to as the domain and the range. For a relation to be classified as a function, each element in the domain must be associated with exactly one element in the range. This is commonly described using the analogy of a vending machine: If you select a particular button for a drink (input), you always get the corresponding drink (output), and not several drinks at once.

In algebraic terms, if we have a set of ordered pairs, like \(\{(-7,-7),(-5,-5),(-3,-3),(0,0)\}\) from the exercise, a function is present only if there are no repeated x-values with different corresponding y-values. In other words, every x-value, which represents an input in the function, maps to one and only one y-value, its output.

Domain and Range

The domain and range of a function are fundamental concepts that tell us about the potential inputs and outputs of a function. In simpler terms, the domain is a set of all the possible x-values that you can put into the function, and the range is the set of all y-values that can come out of the function.

Using our given exercise as an example, the domain consists of all the x-values from our set of ordered pairs, which are \( -7, -5, -3, 0 \). The range similarly consists of corresponding y-values, which in this case, happen to be the same as the domain: \( -7, -5, -3, 0 \). It's worth noting that the domain and range do not need to have the same values; this is just a special case. It is also important to mention that when listing these values, we use curly braces \( \{ \} \) to denote a set, indicating that each member is distinct and order in the set does not matter.

Relation and Function Identification

The process of identifying whether a relation is a function or not is a critical skill in algebra. The exercise presented offers an excellent example. To identify a function, we apply the 'Vertical Line Test'. If you can draw a vertical line that intersects the graph of the relation at more than one point, then the relation is not a function. This is because that would mean a single x-value has multiple y-values associated with it.

However, when working with sets of ordered pairs, like in our example, you can simply inspect the x-values. If each x-value is unique within the set, then it passes the test and is indeed a function. In the exercise, since there are no repeating x-values, the given relation is a function. We improve our understanding by reinforcing the notion that function identification revolves around the uniqueness of the x-values in relation to y-values, emphasizing the one-to-one relationship between the domain (inputs) and the range (outputs).