Finding The Range Of A Function Y = 2x - 5 With Domain {-2, 0, 2, 4}

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In mathematics, understanding the behavior of functions is crucial, and a key aspect of this is determining the range of a function for a specific domain. The range of a function is the set of all possible output values (y-values) that result from using the function's rule on the input values from the domain (x-values). The domain, on the other hand, is the set of all possible input values for which the function is defined. When we are given a function and a specific domain, finding the range involves evaluating the function for each value in the domain. This article will delve into the process of finding the range of a function when the domain is explicitly provided, using the example function y=2x−5y = 2x - 5 with the domain D:{−2,0,2,4}D: \{-2, 0, 2, 4\}. This exploration will not only illustrate the step-by-step method but also emphasize the underlying principles that govern function behavior and range determination. Understanding these concepts is fundamental in various mathematical applications and provides a solid foundation for more advanced topics in calculus and analysis.

Determining the Range: A Step-by-Step Approach

To find the range of the function y=2x−5y = 2x - 5 for the domain D:{−2,0,2,4}D: \{-2, 0, 2, 4\}, we need to substitute each value from the domain into the function and calculate the corresponding y-value. This process will give us the set of all possible output values, which constitutes the range. Let's break this down step by step:

  1. Substitute x = -2 into the function:

    • y=2(−2)−5y = 2(-2) - 5
    • y=−4−5y = -4 - 5
    • y=−9y = -9

    So, when x is -2, the function yields a y-value of -9. This means that -9 is one of the values in the range.

  2. Substitute x = 0 into the function:

    • y=2(0)−5y = 2(0) - 5
    • y=0−5y = 0 - 5
    • y=−5y = -5

    When x is 0, the function gives us a y-value of -5. Thus, -5 is another value in the range.

  3. Substitute x = 2 into the function:

    • y=2(2)−5y = 2(2) - 5
    • y=4−5y = 4 - 5
    • y=−1y = -1

    For x equals 2, the y-value is -1, adding -1 to our range.

  4. Substitute x = 4 into the function:

    • y=2(4)−5y = 2(4) - 5
    • y=8−5y = 8 - 5
    • y=3y = 3

    Finally, when x is 4, the function outputs a y-value of 3, which is the last value in our range for the given domain.

By performing these substitutions, we have found the corresponding y-values for each x-value in the domain. Now, we can collect these y-values to form the range of the function.

Identifying the Range

After substituting each value from the domain D:{−2,0,2,4}D: \{-2, 0, 2, 4\} into the function y=2x−5y = 2x - 5, we obtained the following y-values:

  • When x=−2x = -2, y=−9y = -9.
  • When x=0x = 0, y=−5y = -5.
  • When x=2x = 2, y=−1y = -1.
  • When x=4x = 4, y=3y = 3.

Therefore, the range of the function for the given domain is the set of these y-values: R:{−9,−5,−1,3}R: \{-9, -5, -1, 3\}. This set represents all the possible output values of the function when the input values are restricted to the specified domain. It's important to note that the order of the elements in the range doesn't matter, but it's common practice to list them in ascending order for clarity. The range provides a clear picture of the function's output behavior within the confines of the domain, highlighting the span of y-values that the function can produce. This understanding is crucial for various applications, including graphing functions, solving equations, and modeling real-world scenarios.

Correct Answer and Common Mistakes

Based on our calculations, the correct answer for the range of the function y=2x−5y = 2x - 5 with the domain D:{−2,0,2,4}D: \{-2, 0, 2, 4\} is:

  • C. R:{−9,−5,−1,3}R: \{-9, -5, -1, 3\}

This range accurately reflects the output values we computed by substituting each domain value into the function. It's essential to double-check these calculations to ensure accuracy. However, when dealing with function ranges, several common mistakes can lead to incorrect answers. One frequent error is miscalculating the arithmetic, especially when dealing with negative numbers. For instance, forgetting to correctly apply the order of operations or making sign errors can lead to incorrect y-values. Another common mistake is confusing the domain and range. Students might mistakenly list the domain values as the range or vice versa. To avoid this, it's crucial to remember that the domain consists of the input (x) values, while the range consists of the output (y) values. A third error involves not substituting all the domain values into the function. To determine the range accurately, each value in the domain must be used to calculate the corresponding y-value. Overlooking even one value can lead to an incomplete and incorrect range. Lastly, incorrectly interpreting the function's equation can also cause errors. It's important to understand how the function transforms the input values into output values. For a linear function like y=2x−5y = 2x - 5, each x-value is multiplied by 2, and then 5 is subtracted from the result. Misunderstanding this process can lead to flawed calculations. By being mindful of these common pitfalls and carefully performing each step of the calculation, students can accurately determine the range of a function for a given domain.

Graphical Representation and Interpretation

Visualizing a function through its graph can provide a deeper understanding of its behavior, including its range and domain. The function y=2x−5y = 2x - 5 is a linear function, which means its graph is a straight line. Plotting this line on a coordinate plane allows us to see how the y-values change as the x-values vary. When we restrict the domain to D:{−2,0,2,4}D: \{-2, 0, 2, 4\}, we are essentially looking at specific points on this line corresponding to these x-values. To graph these points, we use the y-values we calculated earlier:

  • For x=−2x = -2, y=−9y = -9, so we have the point (−2,−9)(-2, -9).
  • For x=0x = 0, y=−5y = -5, giving us the point (0,−5)(0, -5).
  • For x=2x = 2, y=−1y = -1, resulting in the point (2,−1)(2, -1).
  • For x=4x = 4, y=3y = 3, which gives us the point (4,3)(4, 3).

Plotting these points on a graph, we can see that they lie on a straight line. The y-coordinates of these points, {−9,−5,−1,3}\{-9, -5, -1, 3\}, represent the range of the function for the given domain. Graphically, the range is the set of all y-values that the function attains within the specified x-values. The graph visually confirms our algebraic calculations and provides an intuitive understanding of the function's behavior. In this case, as x increases within the domain, y also increases, demonstrating a positive slope. The graphical representation is particularly useful for understanding how the function transforms the x-values into y-values and for identifying any patterns or trends. It also helps in distinguishing the domain (x-values) from the range (y-values) and in visualizing the function's output over a specific interval.

Real-World Applications of Range and Domain

Understanding the range and domain of a function is not just a theoretical exercise; it has significant practical applications in various real-world scenarios. Functions are used to model relationships between quantities, and the domain and range provide crucial information about the feasibility and interpretation of these models. For example, consider a function that models the cost of producing a certain number of items. The domain would represent the number of items produced, which cannot be negative, and might have an upper limit due to resource constraints. The range would then represent the possible costs of production, which would also have a lower bound (usually zero) and an upper bound based on the production capacity and cost structure. In physics, functions are used to describe motion, such as the trajectory of a projectile. The domain might represent time, which is non-negative, and the range could represent the height of the projectile, which has a maximum value. Similarly, in economics, supply and demand curves are functions where the domain is the quantity of goods, and the range is the price. The domain and range in these cases provide realistic limits for the model. In computer science, functions are used extensively in algorithms and data structures. The domain could be the size of the input data, and the range could be the execution time or memory usage. Understanding these parameters is crucial for optimizing algorithms and ensuring efficient program performance. In statistics, probability distributions are functions where the domain is the set of possible outcomes, and the range is the probability of each outcome. The range in this case is always between 0 and 1, representing the probabilities. Overall, the concepts of range and domain are fundamental in applying mathematical functions to real-world problems, providing the necessary context for interpreting results and making informed decisions. By understanding these concepts, we can build more accurate and meaningful models of the world around us.

In conclusion, determining the range of a function for a given domain involves substituting each value from the domain into the function and calculating the corresponding output values. For the function y=2x−5y = 2x - 5 and the domain D:{−2,0,2,4}D: \{-2, 0, 2, 4\}, the range is R:{−9,−5,−1,3}R: \{-9, -5, -1, 3\}. Understanding how to find the range is crucial for grasping the behavior of functions and their applications in various fields.