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

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In the realm of mathematics, understanding the behavior of functions is crucial. One key aspect of this understanding lies in determining the range of a function, which represents the set of all possible output values. In this article, we will delve into the process of finding the range of the linear function f(x) = 4x + 9, given the domain D = {-4, -2, 0, 2}. This involves substituting each value from the domain into the function and observing the resulting output values. The collection of these output values constitutes the range of the function for the specified domain. Before we dive into the specifics of this problem, let's first establish a solid foundation by defining what functions, domains, and ranges are.

A function, in mathematical terms, can be visualized as a machine that takes an input, processes it according to a specific rule, and produces an output. This rule is often expressed as an equation, such as f(x) = 4x + 9. The input to this machine is the independent variable, typically denoted as 'x', while the output is the dependent variable, usually represented as 'f(x)' or 'y'. The domain of a function is the set of all permissible input values, the 'x' values that can be plugged into the function without causing any mathematical errors, such as division by zero or taking the square root of a negative number. The range, on the other hand, is the set of all possible output values that the function can produce when given the input values from its domain. It is the collection of 'f(x)' or 'y' values that result from the function's operation on the domain elements. Understanding these fundamental concepts is essential for tackling problems involving functions, including finding their ranges.

Now that we have a clear understanding of the fundamental concepts, let's focus on the specific problem at hand: finding the range of the function f(x) = 4x + 9 given the domain D = {-4, -2, 0, 2}. This function is a linear function, which means that its graph is a straight line. Linear functions are characterized by a constant rate of change, which in this case is represented by the coefficient '4' in front of 'x'. The '+ 9' term represents the y-intercept, the point where the line crosses the y-axis. The domain D = {-4, -2, 0, 2} is a discrete set of values, meaning that it consists of specific, isolated numbers. To find the range, we need to substitute each of these values into the function and calculate the corresponding output. This will give us a set of output values, which will be the range of the function for the given domain. This process is straightforward but requires careful calculation to ensure accuracy. Each substitution provides a data point that contributes to the overall understanding of the function's behavior within the defined domain. By systematically evaluating the function at each domain value, we can construct the range and gain valuable insights into the function's output pattern.

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 will systematically substitute each value from the domain into the function and compute the corresponding output. This process will give us a set of output values that collectively form the range of the function for the specified domain. Let's proceed step-by-step:

  1. Substitute x = -4:

    • f(-4) = 4(-4) + 9
    • f(-4) = -16 + 9
    • f(-4) = -7

    When we substitute -4 into the function, we multiply it by 4, which gives us -16. Then, we add 9 to -16, resulting in -7. This indicates that for the input value of -4, the function outputs -7.

  2. Substitute x = -2:

    • f(-2) = 4(-2) + 9
    • f(-2) = -8 + 9
    • f(-2) = 1

    Substituting -2 into the function, we multiply it by 4 to get -8. Adding 9 to -8 yields 1. This means that when the input is -2, the function's output is 1.

  3. Substitute x = 0:

    • f(0) = 4(0) + 9
    • f(0) = 0 + 9
    • f(0) = 9

    When we substitute 0 into the function, multiplying it by 4 results in 0. Adding 9 to 0 gives us 9. Therefore, for the input value of 0, the function produces an output of 9. This also tells us that the y-intercept of the line represented by this function is 9.

  4. Substitute x = 2:

    • f(2) = 4(2) + 9
    • f(2) = 8 + 9
    • f(2) = 17

    Substituting 2 into the function, we multiply it by 4, which equals 8. Adding 9 to 8 gives us 17. This indicates that when the input is 2, the function outputs 17.

By performing these substitutions, we have obtained the output values corresponding to each input value in the domain. These output values collectively form the range of the function for the given domain. Now, let's gather these values and express the range as a set.

Determining the Correct Range

Now that we have calculated the output values for each element in the domain, we can compile the range of the function. The range is the set of all output values obtained when we substitute the domain values into the function. From our calculations, we have the following:

  • For x = -4, f(x) = -7
  • For x = -2, f(x) = 1
  • For x = 0, f(x) = 9
  • For x = 2, f(x) = 17

Therefore, the range of the function f(x) = 4x + 9 for the domain D = {-4, -2, 0, 2} is the set {-7, 1, 9, 17}. Comparing this result with the given options, we can identify the correct answer.

Let's examine the options:

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, we can clearly see that option A. R = {-7, 1, 9, 17} matches our result. Therefore, option A is the correct answer. The other options contain values that do not correspond to the outputs of the function for the given domain.

This exercise demonstrates the importance of careful calculation and systematic substitution when determining the range of a function. By accurately evaluating the function at each domain value, we can confidently identify the correct range and gain a deeper understanding of the function's behavior.

Conclusion

In conclusion, we have successfully determined the range of the function f(x) = 4x + 9 for the given domain D = {-4, -2, 0, 2}. By systematically substituting each value from the domain into the function and calculating the corresponding output, we found the range to be {-7, 1, 9, 17}. This process highlights the fundamental concept of a function's range and its relationship to the domain. Understanding how to find the range is crucial for analyzing and interpreting the behavior of functions in various mathematical contexts.

This article has provided a detailed, step-by-step approach to solving this particular problem. However, the underlying principles and techniques are applicable to a wide range of function-related problems. By mastering these concepts, students and enthusiasts alike can confidently tackle more complex mathematical challenges. The key takeaways from this exploration include the importance of understanding the definitions of domain and range, the systematic process of substituting domain values into a function, and the careful calculation required to arrive at the correct answer. By applying these principles, one can confidently navigate the world of functions and their ranges, unlocking deeper insights into the fascinating realm of mathematics.

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