Finding The Range Of F(x) = (x-4)(x-2) A Step-by-Step Guide

When delving into the world of mathematical functions, understanding their range is crucial. The range of a function represents the set of all possible output values (y-values) that the function can produce. In this comprehensive article, we will meticulously explore the range of the quadratic function f(x) = (x-4)(x-2). This involves identifying the minimum or maximum value of the function and determining the interval within which all possible output values fall. We will cover the essential steps, including expanding the function, finding the vertex, and using this information to deduce the correct range. By the end of this exploration, you'll not only understand how to find the range for this specific function but also grasp the general principles applicable to quadratic functions. These functions, characterized by their parabolic shape, often appear in various fields of science, engineering, and economics, making their analysis a fundamental skill.

Expanding the Quadratic Function

The first step in determining the range of f(x) = (x-4)(x-2) involves expanding the function into its standard quadratic form. This form, expressed as ax² + bx + c, provides valuable insights into the function's behavior. Expanding the function involves multiplying the binomial factors (x-4) and (x-2) together. We use the distributive property, often remembered by the acronym FOIL (First, Outer, Inner, Last), to ensure that each term in the first binomial is multiplied by each term in the second binomial.

So, let's perform the expansion:

f(x) = (x-4)(x-2) = x(x-2) - 4(x-2) = x² - 2x - 4x + 8 = x² - 6x + 8

The expanded form of the function is f(x) = x² - 6x + 8. This form is critical because it immediately reveals the coefficients a, b, and c, which are essential for further analysis. In this case, a = 1, b = -6, and c = 8. The value of 'a' is particularly significant because it determines the parabola's concavity. Since a = 1, which is positive, the parabola opens upwards. This means the function has a minimum value, which will define the lower bound of the range. Understanding the direction the parabola opens is a foundational element in determining the function's range, as it tells us whether we are looking for a minimum or a maximum point. Furthermore, the expanded form allows us to apply the vertex formula, a key tool in identifying the function's extreme point and, consequently, its range.

Finding the Vertex of the Parabola

The vertex of a parabola is the point where the function reaches its minimum or maximum value. For a quadratic function in the form f(x) = ax² + bx + c, the x-coordinate of the vertex can be found using the formula x = -b / 2a. This formula is derived from completing the square or using calculus, and it provides a straightforward method for locating the parabola's central point.

In our case, f(x) = x² - 6x + 8, where a = 1 and b = -6. Plugging these values into the formula, we get:

x = -(-6) / (2 * 1) = 6 / 2 = 3

Thus, the x-coordinate of the vertex is 3. To find the y-coordinate, which represents the minimum value of the function, we substitute x = 3 back into the expanded function:

f(3) = (3)² - 6(3) + 8 = 9 - 18 + 8 = -1

Therefore, the vertex of the parabola is (3, -1). This point is the lowest point on the graph because the parabola opens upwards. The y-coordinate of the vertex, -1, is the minimum value of the function. Knowing the vertex is pivotal in determining the range because it provides the lower (or upper) bound of all possible output values. In this case, since the parabola opens upwards, the y-coordinate of the vertex is the minimum y-value, and the range will include all y-values greater than or equal to this minimum. The vertex calculation is not just a mathematical step; it's a critical point in understanding the function's overall behavior and its range.

Determining the Range

Now that we have identified the vertex of the parabola as (3, -1), we can determine the range of the function f(x) = (x-4)(x-2). The vertex represents the minimum point of the function because the coefficient of the x² term (a) is positive, indicating that the parabola opens upwards. This means that the y-coordinate of the vertex, which is -1, is the minimum y-value that the function can achieve.

Since the parabola opens upwards, the function's values will extend infinitely upwards from this minimum point. Therefore, the range of the function includes all real numbers greater than or equal to -1. This can be expressed in several ways: using inequality notation as y ≥ -1, in interval notation as [-1, ∞), or in set notation as {y | y is a real number and y ≥ -1}.

Understanding that the parabola opens upwards and that the vertex is the minimum point is crucial for correctly determining the range. If the parabola opened downwards (i.e., if 'a' were negative), the vertex would represent the maximum point, and the range would include all real numbers less than or equal to the y-coordinate of the vertex. Visualizing the graph of the function can be extremely helpful in confirming the range. A quick sketch or using graphing software can illustrate the parabola's shape and its orientation, making it clear that all y-values above -1 are included in the range. This final step of range determination is a culmination of the previous steps, relying on the accurate expansion of the function and the precise calculation of the vertex.

Conclusion: The Range of f(x) = (x-4)(x-2)

In conclusion, the range of the quadratic function f(x) = (x-4)(x-2) is all real numbers greater than or equal to -1. We arrived at this conclusion by first expanding the function to its standard quadratic form, f(x) = x² - 6x + 8. This allowed us to identify the coefficients necessary for finding the vertex. By using the vertex formula, we determined that the vertex of the parabola is (3, -1). Since the coefficient of the x² term is positive, the parabola opens upwards, meaning the vertex represents the minimum point of the function.

The y-coordinate of the vertex, -1, is therefore the minimum y-value in the range. Because the parabola extends upwards indefinitely, the range includes all y-values greater than or equal to -1. This understanding is not only crucial for this specific problem but also for analyzing other quadratic functions. The process of expanding the function, finding the vertex, and considering the parabola's orientation is a universally applicable method for determining the range of quadratic functions. Understanding the range of functions is a fundamental concept in mathematics, with applications in various fields. This detailed exploration provides a solid foundation for tackling similar problems and further delving into the properties of quadratic functions. By mastering this process, students and practitioners can confidently analyze and interpret the behavior of these essential mathematical constructs. The range determination is a key element in the comprehensive understanding of quadratic functions.

Correct Answer

The correct answer is B. all real numbers greater than or equal to -1