Is {(5, 9), (0, 7), (5, 10)} A Function? Determine Domain And Range

Understanding Relations and Functions

In the realm of mathematics, the concept of a function is fundamental. To grasp this concept fully, we first need to understand what a relation is. A relation, in its simplest form, is a set of ordered pairs. These ordered pairs link two sets of elements. The first set is known as the domain, and the second set is known as the range. The domain represents the set of all possible input values, while the range represents the set of all possible output values. Think of it like a machine: you input something (from the domain), and the machine processes it and gives you an output (from the range).

Now, not every relation qualifies as a function. A function is a special type of relation with a crucial rule: each element in the domain can only be associated with one unique element in the range. In simpler terms, for every input, there can be only one output. This is the golden rule of functions. It ensures that the function behaves predictably and consistently. To illustrate, consider a vending machine. If you press the button for a specific soda (the input), you expect to get that specific soda (the output) every time. If the machine sometimes dispensed a different soda for the same button press, it wouldn't be functioning as a function should. This uniqueness of output for each input is what sets functions apart from general relations.

To further clarify, let's consider some examples. Imagine a relation that pairs students with their favorite subjects. If one student has multiple favorite subjects, this relation is not a function because one input (the student) has multiple outputs (favorite subjects). However, if each student has only one favorite subject, then this relation is a function. Another example could be a relation pairing employees with their salaries. If each employee has only one salary, this is a function. But if an employee somehow had two different salaries recorded, it would not be a function. These examples underscore the importance of the one-to-one or many-to-one mapping from the domain to the range for a relation to be considered a function. Understanding this distinction is key to solving problems like the one presented, where we need to analyze a given relation and determine if it meets the criteria to be a function.

Analyzing the Given Relation: {(5, 9), (0, 7), (5, 10)}

To determine whether the given relation, {(5, 9), (0, 7), (5, 10)}, defines a function, we must carefully examine the ordered pairs and apply the fundamental definition of a function. Remember, a function requires that each element in the domain (the first value in the ordered pair) maps to only one unique element in the range (the second value in the ordered pair). In other words, no input can have multiple outputs.

Let's break down the given relation. We have three ordered pairs: (5, 9), (0, 7), and (5, 10). The domain consists of the first elements of each pair, which are 5 and 0. The range consists of the second elements, which are 9, 7, and 10. Now, let's focus on the crucial aspect: does each element in the domain map to a unique element in the range? We can see that the element 5 in the domain appears in two ordered pairs: (5, 9) and (5, 10). This means that the input 5 is associated with two different outputs, 9 and 10. This immediately violates the definition of a function.

Since the input 5 has two different outputs, we can definitively conclude that the given relation is not a function. The presence of even one input mapping to multiple outputs is sufficient to disqualify a relation from being a function. This is a critical point to remember when analyzing relations. To further solidify this understanding, imagine trying to graph this relation. The points (5, 9) and (5, 10) would be vertically aligned on the coordinate plane. This vertical alignment is a visual cue that the relation is not a function, as it fails the vertical line test (a graphical method to determine if a relation is a function; if a vertical line intersects the graph at more than one point, it is not a function).

Therefore, the existence of the ordered pairs (5, 9) and (5, 10) within the relation is the decisive factor in determining that it is not a function. This underscores the importance of the uniqueness criterion in the definition of a function. In summary, the given relation fails to meet the required condition for functionality because the input 5 maps to two different outputs, making it a relation but not a function.

Identifying the Domain and Range

Even though the given relation, {(5, 9), (0, 7), (5, 10)}, is not a function, we can still identify its domain and range. Understanding the domain and range is crucial for describing the scope and behavior of any relation, whether it's a function or not. The domain, as mentioned earlier, is the set of all possible input values, which are the first elements in the ordered pairs. The range, on the other hand, is the set of all possible output values, which are the second elements in the ordered pairs.

To determine the domain of the relation {(5, 9), (0, 7), (5, 10)}, we simply list all the first elements present in the ordered pairs. These elements are 5, 0, and 5. However, in set notation, we only list unique elements, so we don't repeat the 5. Therefore, the domain of this relation is {0, 5}. This means that the relation accepts 0 and 5 as valid inputs. The order in which we list the elements in a set does not matter; the set {5, 0} is equivalent to {0, 5}.

Next, let's identify the range of the relation. To do this, we list all the second elements present in the ordered pairs. These elements are 9, 7, and 10. Since all these values are unique, we include each of them in the range. Thus, the range of the relation is {7, 9, 10}. This indicates that the relation produces the values 7, 9, and 10 as possible outputs. Like the domain, the order of elements in the range doesn't matter; {10, 7, 9} represents the same range.

In summary, for the relation {(5, 9), (0, 7), (5, 10)}, the domain is {0, 5}, and the range is {7, 9, 10}. It's important to note that determining the domain and range is a separate process from determining whether a relation is a function. We can always identify the domain and range of a relation, but the relation is a function only if it satisfies the uniqueness criterion – each input mapping to only one output. In this case, while we can identify the domain and range, the relation fails the function test due to the input 5 having two outputs.

Conclusion

In conclusion, when analyzing the relation {(5, 9), (0, 7), (5, 10)}, we've determined that it does not define a function. This is because the input value 5 maps to two different output values, 9 and 10, which violates the fundamental requirement that each element in the domain must map to only one element in the range for a relation to be considered a function. Understanding this principle is crucial for differentiating functions from general relations in mathematics.

Despite the relation not being a function, we were able to successfully identify its domain and range. The domain, which represents the set of all possible input values, is {0, 5}. The range, representing the set of all possible output values, is {7, 9, 10}. Identifying the domain and range provides valuable information about the relation's scope and the values it can handle and produce, even if it doesn't meet the criteria for a function.

This exercise underscores the importance of carefully examining ordered pairs and applying the definition of a function when presented with a relation. It also highlights that the domain and range are properties inherent to any relation, regardless of whether it's a function. By mastering these concepts, you'll be well-equipped to analyze and classify various mathematical relations and functions, which is a fundamental skill in many areas of mathematics and its applications.