Finding Domain And Range Of A Function Given Ordered Pairs

In mathematics, understanding the domain and range of a function is crucial for analyzing its behavior and characteristics. This article provides a comprehensive guide on how to determine the domain and range of a function, particularly when the function is presented as a set of ordered pairs. We will walk through the process step-by-step, using the example function: $\left\{\left(-\frac{3}{2}, \frac{5}{4}\right),\left(-\frac{1}{2},-\frac{3}{4}\right),\left(\frac{1}{2},-\frac{3}{4}\right),\left(\frac{3}{2}, \frac{5}{4}\right),\left(\frac{5}{2}, \frac{21}{4}\right)\right\}$.

Understanding Domain and Range

Before diving into the solution, let's clarify the fundamental concepts of domain and range. The domain of a function is the set of all possible input values (often denoted as 'x') for which the function is defined. In simpler terms, it's the collection of all 'x' values that you can plug into the function and get a valid output. The range, on the other hand, is the set of all possible output values (often denoted as 'y') that the function can produce. It represents the collection of all 'y' values that result from applying the function to the values in its domain.

When a function is given as a set of ordered pairs, each pair is in the form (x, y), where 'x' belongs to the domain and 'y' belongs to the range. To find the domain, we simply collect all the 'x' values from the ordered pairs. Similarly, to find the range, we collect all the 'y' values. It's important to remember that sets do not contain duplicate elements, so if a value appears multiple times, we only include it once in the domain or range.

In our specific example, the function is defined as a set of ordered pairs: $\left\{\left(-\frac{3}{2}, \frac{5}{4}\right),\left(-\frac{1}{2},-\frac{3}{4}\right),\left(\frac{1}{2},-\frac{3}{4}\right),\left(\frac{3}{2}, \frac{5}{4}\right),\left(\frac{5}{2}, \frac{21}{4}\right)\right\}$. Our task is to identify all the 'x' values to determine the domain and all the 'y' values to determine the range.

Identifying the Domain

The domain is the set of all first elements (x-coordinates) in the ordered pairs. Let's extract these values from the given set:

• From the pair $\left(-\frac{3}{2}, \frac{5}{4}\right)$, we have $-\frac{3}{2}$.
• From the pair $\left(-\frac{1}{2},-\frac{3}{4}\right)$, we have $-\frac{1}{2}$.
• From the pair $\left(\frac{1}{2},-\frac{3}{4}\right)$, we have $\frac{1}{2}$.
• From the pair $\left(\frac{3}{2}, \frac{5}{4}\right)$, we have $\frac{3}{2}$.
• From the pair $\left(\frac{5}{2}, \frac{21}{4}\right)$, we have $\frac{5}{2}$.

Now, we collect these values into a set, ensuring that we don't include any duplicates. The domain of the function is:

$\left\{-\frac{3}{2}, -\frac{1}{2}, \frac{1}{2}, \frac{3}{2}, \frac{5}{2}\right\}$

This set represents all the possible input values for the function. We can see that the function is defined for five distinct x-values.

Determining the Range

Next, we need to find the range, which is the set of all second elements (y-coordinates) in the ordered pairs. Let's extract these values from the given set:

• From the pair $\left(-\frac{3}{2}, \frac{5}{4}\right)$, we have $\frac{5}{4}$.
• From the pair $\left(-\frac{1}{2},-\frac{3}{4}\right)$, we have $-\frac{3}{4}$.
• From the pair $\left(\frac{1}{2},-\frac{3}{4}\right)$, we have $-\frac{3}{4}$.
• From the pair $\left(\frac{3}{2}, \frac{5}{4}\right)$, we have $\frac{5}{4}$.
• From the pair $\left(\frac{5}{2}, \frac{21}{4}\right)$, we have $\frac{21}{4}$.

We collect these values into a set, and since sets do not allow duplicates, we only include each unique value once. The range of the function is:

$\left\{-\frac{3}{4}, \frac{5}{4}, \frac{21}{4}\right\}$

This set represents all the possible output values for the function. We can see that the function produces three distinct y-values.

Step-by-Step Solution

To summarize, let's break down the process into clear steps:

1. Identify the Ordered Pairs: Begin by clearly identifying the set of ordered pairs that define the function. In our case, the function is given as: $\left\{\left(-\frac{3}{2}, \frac{5}{4}\right),\left(-\frac{1}{2},-\frac{3}{4}\right),\left(\frac{1}{2},-\frac{3}{4}\right),\left(\frac{3}{2}, \frac{5}{4}\right),\left(\frac{5}{2}, \frac{21}{4}\right)\right\}$.
2. Extract the x-coordinates: Collect all the first elements (x-coordinates) from each ordered pair. These values will form the domain. In our example, the x-coordinates are: $-\frac{3}{2}$, $-\frac{1}{2}$, $\frac{1}{2}$, $\frac{3}{2}$, and $\frac{5}{2}$.
3. Form the Domain Set: Create a set using the extracted x-coordinates, ensuring that you do not include any duplicate values. The domain for our function is: $\left\{-\frac{3}{2}, -\frac{1}{2}, \frac{1}{2}, \frac{3}{2}, \frac{5}{2}\right\}$.
4. Extract the y-coordinates: Collect all the second elements (y-coordinates) from each ordered pair. These values will form the range. In our example, the y-coordinates are: $\frac{5}{4}$, $-\frac{3}{4}$, $-\frac{3}{4}$, $\frac{5}{4}$, and $\frac{21}{4}$.
5. Form the Range Set: Create a set using the extracted y-coordinates, ensuring that you do not include any duplicate values. The range for our function is: $\left\{-\frac{3}{4}, \frac{5}{4}, \frac{21}{4}\right\}$.

By following these steps, you can systematically determine the domain and range of any function presented as a set of ordered pairs.

Generalizing the Concept

While this article focuses on functions defined by a set of ordered pairs, the concept of domain and range extends to functions defined in other ways, such as equations or graphs. For functions defined by equations, the domain is often restricted by values that would lead to undefined operations, such as division by zero or taking the square root of a negative number. The range can be determined by analyzing the behavior of the function and identifying the possible output values.

For functions represented graphically, the domain can be visualized as the set of all x-values covered by the graph, and the range can be visualized as the set of all y-values covered by the graph. Understanding how to determine the domain and range in different representations is essential for a comprehensive understanding of functions.

Importance of Domain and Range

The domain and range are fundamental concepts in mathematics because they provide crucial information about the behavior and limitations of a function. Knowing the domain helps us understand what input values are permissible, preventing us from attempting to evaluate the function at points where it is not defined. This is particularly important in real-world applications where input values might represent physical quantities with inherent constraints.

The range, on the other hand, tells us the possible output values of the function. This information is valuable for understanding the function's behavior and its potential applications. For example, if a function models the height of a projectile, the range would tell us the maximum height the projectile can reach.

Furthermore, the domain and range are essential for comparing and classifying functions. Functions with different domains or ranges may exhibit different properties and behaviors. Understanding these differences allows mathematicians and scientists to choose the appropriate functions for specific applications.

In calculus, the domain and range play a critical role in determining the continuity and differentiability of a function. A function must be defined at a point to be continuous or differentiable at that point, so the domain is a prerequisite for these concepts. The range, in turn, helps us understand the function's extrema (maximum and minimum values) and its overall behavior.

Conclusion

In conclusion, finding the domain and range of a function is a fundamental skill in mathematics. For functions presented as a set of ordered pairs, the domain is the set of all x-coordinates, and the range is the set of all y-coordinates. By systematically extracting these values and forming sets, we can easily determine the domain and range. This understanding is crucial for analyzing the behavior of functions and their applications in various fields. The example function $\left\{\left(-\frac{3}{2}, \frac{5}{4}\right),\left(-\frac{1}{2},-\frac{3}{4}\right),\left(\frac{1}{2},-\frac{3}{4}\right),\left(\frac{3}{2}, \frac{5}{4}\right),\left(\frac{5}{2}, \frac{21}{4}\right)\right\}$ has a domain of $\left\{-\frac{3}{2}, -\frac{1}{2}, \frac{1}{2}, \frac{3}{2}, \frac{5}{2}\right\}$ and a range of $\left\{-\frac{3}{4}, \frac{5}{4}, \frac{21}{4}\right\}$. This step-by-step approach can be applied to any function defined as a set of ordered pairs, providing a solid foundation for further mathematical exploration.