Domain & Range: F(x) = -(x+3)(x-1) Explained

Let's dive into understanding the domain and range of the quadratic function f(x) = -(x+3)(x-1). This is a fundamental concept in mathematics, particularly when analyzing functions and their graphical representations. The domain refers to all possible input values (x-values) that the function can accept, while the range refers to all possible output values (y-values) that the function can produce. For any function, determining these two aspects provides a comprehensive understanding of its behavior and limitations. In the context of our given quadratic function, understanding its domain and range will involve looking at the graph, identifying key features, and applying algebraic principles.

Analyzing the Function f(x) = -(x+3)(x-1)

To kick things off, let's take a closer look at the function f(x) = -(x+3)(x-1). This is a quadratic function expressed in factored form. Expanding this gives us f(x) = -x² - 2x + 3. Quadratic functions are characterized by their parabolic shape when graphed, either opening upwards or downwards. In this case, the negative coefficient in front of the x² term indicates that the parabola opens downwards. This is crucial because it tells us that the function has a maximum value. Finding this maximum value, as well as understanding the end behavior of the parabola, will be essential in determining the range of the function. Additionally, we must consider if there are any restrictions on the x-values that can be plugged into the function. For polynomial functions like this, the domain is generally all real numbers since there are no denominators, square roots, or logarithms that could cause undefined values. Thus, our focus will be on accurately determining the range based on the parabola's orientation and vertex.

Determining the Domain

The domain of a function is the set of all possible input values (x-values) for which the function is defined. For the given function f(x) = -(x+3)(x-1), which is a polynomial (specifically, a quadratic), there are no restrictions on the values that x can take. Polynomial functions are defined for all real numbers because you can substitute any real number into the function and obtain a real number as the output. There are no denominators that could be zero, no square roots of negative numbers, and no logarithms of non-positive numbers to worry about. Therefore, the domain of f(x) = -(x+3)(x-1) is all real numbers. In interval notation, this is represented as (-∞, ∞). Understanding this aspect of the function helps to establish a complete picture before moving on to the range, which will rely on the specific characteristics of the parabola defined by the quadratic expression.

Finding the Range

The range of a function is the set of all possible output values (y-values) that the function can produce. Since f(x) = -(x+3)(x-1) is a downward-opening parabola, it has a maximum value. To find this maximum value, we first need to find the vertex of the parabola. The x-coordinate of the vertex can be found using the formula x = -b / 2a, where a and b are the coefficients in the quadratic equation f(x) = ax² + bx + c. In our case, after expanding the original function, we have f(x) = -x² - 2x + 3, so a = -1 and b = -2. Plugging these values into the formula, we get x = -(-2) / (2 * -1) = 1 / -1 = -1. Now, we substitute x = -1 back into the function to find the y-coordinate of the vertex: f(-1) = -(-1 + 3)(-1 - 1) = -(2)(-2) = 4. This means the vertex of the parabola is at the point (-1, 4). Since the parabola opens downwards, the maximum value of the function is 4. Therefore, the range of the function is all real numbers less than or equal to 4. In interval notation, this is represented as (-∞, 4]. This carefully determined range completes our understanding of the function's output possibilities.

Analyzing the Options

Now let's review the given options based on our analysis:

A. The domain is all real numbers less than or equal to 4, and the range is all real numbers such that -3 ≤ x ≤ 1. B. [The original options are missing, so this part cannot be completed accurately.]

Based on our findings, option A is incorrect. The domain is all real numbers, not just those less than or equal to 4. The range is all real numbers less than or equal to 4, but the description in option A is not correct either. Without the other options, we cannot definitively choose the correct one. However, we have established the correct domain and range through our analysis. The accurate description should state that the domain is all real numbers and the range is all real numbers less than or equal to 4.

Conclusion

In summary, the domain of the function f(x) = -(x+3)(x-1) is all real numbers, and the range is all real numbers less than or equal to 4. Understanding how to determine the domain and range of functions is crucial in mathematics, providing a foundation for further analysis and applications. Remember, the domain focuses on the possible input values, while the range focuses on the resulting output values. For quadratic functions, identifying the vertex and the direction of the parabola is key to finding the range. By combining algebraic methods with graphical understanding, we can accurately describe the behavior of these functions.

For more information on functions and their domains and ranges, you can visit Khan Academy's Functions and Domain/Range section.