Finding The Range Of F(x) = 4x + 9 With Domain {-4, -2, 0, 2}

Determining the range of a function is a fundamental concept in mathematics, especially when dealing with functions defined over a specific domain. The range represents the set of all possible output values that the function can produce when given inputs from its domain. This article delves into how to find the range of the linear function $f(x) = 4x + 9$ when its domain is restricted to a finite set $D = \{-4, -2, 0, 2\}$. By systematically evaluating the function at each point in the domain, we can accurately identify the range. Understanding this process is crucial not only for solving mathematical problems but also for grasping the behavior of functions in various applications.

The process of finding the range involves substituting each element of the domain into the function and calculating the corresponding output. Each calculation provides a value that is part of the range. After evaluating the function for all elements in the domain, we compile the resulting set of output values, ensuring that we only include unique values. This set of unique output values constitutes the range of the function for the given domain. In this article, we will walk through each step of this process, providing a clear and concise explanation to ensure a solid understanding of how to determine the range of a function.

Step-by-Step Calculation of the Range

To determine the range of the function $f(x) = 4x + 9$ for the domain $D = \{-4, -2, 0, 2\}$, we need to evaluate the function at each $x$-value in the domain. This involves substituting each element of the domain into the function and calculating the corresponding $f(x)$ value. Let's go through each calculation step-by-step:

1. Evaluate f(x) at x = -4:

   • Substitute $x = -4$ into the function: $f(-4) = 4(-4) + 9$.
   • Perform the multiplication: $4(-4) = -16$.
   • Add 9 to the result: $-16 + 9 = -7$.
   • Thus, $f(-4) = -7$.

2. Evaluate f(x) at x = -2:

   • Substitute $x = -2$ into the function: $f(-2) = 4(-2) + 9$.
   • Perform the multiplication: $4(-2) = -8$.
   • Add 9 to the result: $-8 + 9 = 1$.
   • Thus, $f(-2) = 1$.

3. Evaluate f(x) at x = 0:

   • Substitute $x = 0$ into the function: $f(0) = 4(0) + 9$.
   • Perform the multiplication: $4(0) = 0$.
   • Add 9 to the result: $0 + 9 = 9$.
   • Thus, $f(0) = 9$.

4. Evaluate f(x) at x = 2:

   • Substitute $x = 2$ into the function: $f(2) = 4(2) + 9$.
   • Perform the multiplication: $4(2) = 8$.
   • Add 9 to the result: $8 + 9 = 17$.
   • Thus, $f(2) = 17$.

By evaluating the function at each point in the domain, we have found the corresponding $f(x)$ values: $-7$, $1$, $9$, and $17$. These values form the range of the function for the given domain. Therefore, the range $R$ is the set of these values.

Identifying the Correct Range

Now that we have calculated the output values for each element in the domain, we can assemble the range $R$. The range consists of the set of all $f(x)$ values we found in the previous step. In this case, the values are $-7$, $1$, $9$, and $17$. Thus, the range $R$ is the set $\{-7, 1, 9, 17\}$.

Let's analyze the given options to determine which one matches our calculated range:

A. $R = \{-7, 1, 9, 17\}$ B. $R = \{-7, -1, 9, 17\}$ C. $R = \{-17, -9, -1, 17\}$ D. $R = \{1, 7, 9, 17\}$

By comparing our calculated range with the options, it's clear that option A, $R = \{-7, 1, 9, 17\}$, matches the values we computed. This confirms that option A is the correct answer. The other options contain values that do not correspond to the output of the function for the given domain.

The Importance of Domain in Determining Range

Understanding the role of the domain in determining the range is crucial for mastering function analysis. The domain of a function is the set of all possible input values (often $x$-values) for which the function is defined. The range, on the other hand, is the set of all possible output values (often $y$-values or $f(x)$ values) that the function produces when these input values are used. The domain acts as a constraint on the function, dictating which inputs are permissible and thereby influencing the possible outputs.

When the domain is restricted, as in the example $D = \{-4, -2, 0, 2\}$, the range is similarly constrained. Only the outputs corresponding to these specific inputs are included in the range. This is different from the range of the function over all real numbers, where the function could potentially produce a much wider set of output values. For example, consider the linear function $f(x) = 4x + 9$. Over the entire set of real numbers, its range would also be the set of all real numbers. However, with the restricted domain $D$, the range is limited to the values obtained by substituting the elements of $D$ into $f(x)$.

This concept is particularly important in practical applications where functions are used to model real-world phenomena. In many cases, the inputs to a function are naturally restricted due to physical or contextual limitations. For instance, if a function models the height of a projectile over time, the domain might be restricted to non-negative values since time cannot be negative. Similarly, if a function models the number of items sold, the domain would likely be restricted to non-negative integers. In each of these scenarios, the restricted domain directly affects the possible range of outputs, providing a more realistic and meaningful representation of the situation.

Common Mistakes to Avoid

When determining the range of a function, it's important to avoid common mistakes that can lead to incorrect results. Here are a few key pitfalls to watch out for:

1. Forgetting to Evaluate All Domain Elements:

   • A common mistake is only evaluating the function at some, but not all, elements of the domain. To accurately determine the range, you must substitute every value from the domain into the function. Omitting even one value can lead to an incomplete or incorrect range.

2. Including Values Not in the Domain:

   • The range is determined solely by the outputs produced by the inputs within the domain. Do not consider any other $x$-values or their corresponding $f(x)$ values. Including values from outside the domain will result in an incorrect range.

3. Incorrect Arithmetic:

   • Careless arithmetic errors when substituting domain values and calculating $f(x)$ can lead to wrong answers. Double-check each calculation to ensure accuracy. This is especially important when dealing with negative numbers, fractions, or more complex expressions.

4. Misunderstanding the Definition of Range:

   • The range is the set of output values ($f(x)$ values), not the input values ($x$ values). Confusing the range with the domain is a fundamental error. Always remember to substitute domain values into the function to find the corresponding range values.

5. Not Recognizing Duplicates:

   • The range is a set of unique values. If the same $f(x)$ value is produced by multiple elements in the domain, it should only be included once in the range. Failing to eliminate duplicates can lead to an incorrect representation of the range.

By being mindful of these common mistakes and carefully following the steps outlined in this article, you can accurately determine the range of a function for a given domain.

Practical Applications of Understanding Range

Understanding the range of a function is not just an abstract mathematical concept; it has numerous practical applications across various fields. The range provides critical insights into the possible outcomes or limitations of a system or model represented by the function. Here are a few examples:

1. Physics and Engineering:

   • In physics, functions are used to model physical phenomena such as projectile motion, electrical circuits, and thermodynamics. The range of these functions can represent the possible values of physical quantities like height, voltage, temperature, or velocity. For instance, in projectile motion, the range of the height function indicates the maximum height the projectile can reach. Similarly, in electrical circuits, the range of a current function shows the possible current values that can flow through the circuit. Understanding these ranges helps engineers design systems that operate within safe and efficient limits.

2. Economics and Finance:

   • Economic and financial models often use functions to represent relationships between variables such as supply and demand, cost and revenue, or investment returns over time. The range of these functions can represent possible profit levels, market prices, or investment outcomes. For example, the range of a profit function might indicate the maximum and minimum profits a company can expect under certain conditions. In finance, the range of an investment return function can help investors assess the potential risk and reward associated with different investment options.

3. Computer Science:

   • In computer science, functions are used extensively in algorithms and software development. The range of a function can represent the set of possible output values or the computational complexity of an algorithm. For example, the range of a sorting algorithm might indicate the number of steps required to sort a list of items under different conditions. Understanding the range helps in designing efficient algorithms and reliable software systems.

4. Data Analysis and Statistics:

   • In data analysis and statistics, functions are used to model and analyze data sets. The range of a function can represent the spread or variability of data points. For example, in a regression analysis, the range of the predicted values indicates the possible outcomes based on the model. Understanding the range helps in interpreting data, making predictions, and drawing conclusions.

5. Everyday Life:

   • Even in everyday life, understanding the range of a function can be helpful. For example, if you are planning a trip, you might use a function to estimate the total cost based on the distance traveled and the price of gas. The range of this function would give you an idea of the possible costs you might incur. Similarly, if you are tracking your fitness progress, you might use a function to model your weight loss or gain over time. The range of this function would indicate the possible outcomes of your fitness plan.

Conclusion

In conclusion, determining the range of a function for a given domain is a crucial skill in mathematics with wide-ranging applications. By systematically evaluating the function at each point in the domain, we can accurately identify the set of all possible output values. In the specific example of the function $f(x) = 4x + 9$ with the domain $D = \{-4, -2, 0, 2\}$, we found the range to be $R = \{-7, 1, 9, 17\}$. This process highlights the importance of understanding the relationship between the domain and range and the practical implications of this concept in various fields. By avoiding common mistakes and applying the step-by-step approach outlined in this article, you can confidently tackle problems involving function ranges and appreciate their significance in real-world scenarios.