In mathematics, understanding the range of a function is critical to analyzing how a function behaves based on its output values. The range refers to all possible values a function can output when different inputs from its domain are applied. For the given function, expressed as a set of ordered pairs {(–2, 0), (–4, –3), (2, –9), (0, 5), (–5, 7)}, the task is to determine the range.
This article explores the question: what is the range of the given function? {(–2, 0), (–4, –3), (2, –9), (0, 5), (–5, 7)}. We will break down the concept of the range in functions, guide you through the steps to calculate the range, and provide examples to help you fully grasp the concept. Whether you are new to functions or seeking a deeper understanding, this guide will equip you with everything you need to find the range of a function.
What is the Range of a Function?
The range of a function refers to all the possible output values (y-values) that a function can produce. In the context of ordered pairs, each pair is made up of an input (x-value) and an output (y-value). The range is derived by looking at the set of all output values across the given pairs.
For the given function:
{(–2, 0), (–4, –3), (2, –9), (0, 5), (–5, 7)},
the range can be determined by identifying all of the y-values: 0, –3, –9, 5, 7. Therefore, the range of this function is {0, –3, –9, 5, 7}.
The range is important because it tells us the possible values the function can output based on its domain (x-values). In this example, we know that when the function is applied to the x-values, it will output one of the values within the range.
How to Determine the Range of a Function?
Step-by-Step Approach
To determine the range of a function, follow these simple steps:
- Identify the Ordered Pairs
Start by identifying the set of ordered pairs. In this case, the given function is {(–2, 0), (–4, –3), (2, –9), (0, 5), (–5, 7)}. - Extract the Y-Values
The next step is to extract the y-values from each pair. The y-values in this function are 0, –3, –9, 5, 7. - List the Y-Values
Once you have extracted the y-values, list them in a set. Ensure no value is repeated. The range of the function is {0, –3, –9, 5, 7}. - Double-Check for Completeness
Ensure that all y-values from the ordered pairs have been included in the range.
By following these steps, you can easily determine the range of any function given in the form of ordered pairs.
Why is the Range Important in Understanding a Function?
The range of a function provides critical insights into the behavior of the function. It tells us the set of possible outcomes or outputs for the function’s inputs. Understanding the range is particularly important for:
- Analyzing Function Behavior: The range helps in understanding how a function behaves over its domain. For example, if the range is restricted, it tells us that the function cannot produce certain values.
- Graphing Functions: When graphing a function, knowing the range allows us to know the extent of the y-axis values.
- Real-World Applications: In practical applications, the range can represent real-world constraints. For example, the range of a function modeling temperature might indicate the possible temperatures an environment can reach.
By understanding the range, you get a fuller picture of how the function operates across its domain.
Common Questions About Function Range
1. What is the range of a set of ordered pairs?
The range is the set of y-values in a set of ordered pairs. For example, in the function {(–2, 0), (–4, –3), (2, –9), (0, 5), (–5, 7)}, the range is {0, –3, –9, 5, 7}.
2. How do I determine the range of a function?
To determine the range, extract the y-values from each ordered pair and list them as a set. Ensure no value is repeated.
3. Why is it important to understand the range of a function?
The range helps in analyzing the possible outputs of a function and understanding its behavior over a domain.
4. Can a function have an infinite range?
Yes, some functions can have an infinite range, especially those that continue without bound. However, in this example, the range is finite.
5. How does the range differ from the domain?
While the range refers to the set of possible outputs (y-values), the domain refers to the set of all possible inputs (x-values).
The Role of the Domain in Determining the Range
The domain and range of a function are closely linked. The domain refers to all the possible input values (x-values) that can be plugged into the function, while the range is the resulting set of output values (y-values). In the case of the given function:
{(–2, 0), (–4, –3), (2, –9), (0, 5), (–5, 7)}:
- The domain consists of the x-values: –2, –4, 2, 0, –5.
- The range consists of the corresponding y-values: 0, –3, –9, 5, 7.
Both domain and range help us understand how the function maps inputs to outputs and what values are possible within the function’s framework.
Conclusion
In summary, the range of the given function {(–2, 0), (–4, –3), (2, –9), (0, 5), (–5, 7)} is {0, –3, –9, 5, 7}. By analyzing the set of ordered pairs, we can easily extract the y-values to determine the range. Understanding the range is essential for analyzing the behavior of the function and interpreting its possible outputs. Whether you’re dealing with a simple function or more complex cases, determining the range follows a similar process of identifying the y-values in the ordered pairs.
FAQ’s
Q. What is the range of the function {(–2, 0), (–4, –3), (2, –9), (0, 5), (–5, 7)}?
A. The range is {0, –3, –9, 5, 7}, representing the set of y-values.
Q. How do you find the range of a function given ordered pairs?
A. Extract the y-values from the ordered pairs and list them as a set to find the range.
Q. What does the range of a function represent?
A. The range represents the set of all possible outputs (y-values) that the function can produce.
Q. Can the range of a function be infinite?
A. Yes, some functions can have an infinite range, but in this example, the range is finite.
Q. How are domain and range related in a function?
A. The domain refers to the set of possible inputs (x-values), while the range refers to the possible outputs (y-values). Both are essential to fully understanding a function’s behavior.
