Domain And Range Of F(x) = -(x+3)(x-1) A Comprehensive Guide

Before diving into the specifics of the function f(x) = -(x+3)(x-1), let's first clarify the concepts of domain and range. These are fundamental ideas in mathematics, particularly when dealing with functions. Understanding them is crucial for analyzing and interpreting the behavior of functions.

The domain of a function is essentially the set of all possible input values (often represented as 'x') for which the function is defined. Think of it as the set of all 'x' values that you can plug into the function without causing any mathematical errors or undefined results. For instance, if a function involves a square root, the domain would exclude any 'x' values that result in a negative number under the radical, as the square root of a negative number is not a real number. Similarly, if a function has a denominator, the domain would exclude any 'x' values that make the denominator zero, as division by zero is undefined.

For polynomial functions, like the one we are examining, the domain is typically all real numbers. This is because you can substitute any real number for 'x' in a polynomial expression, and you will always get a valid real number output. There are no restrictions imposed by square roots, denominators, or other potentially problematic operations.

On the other hand, the range of a function is the set of all possible output values (often represented as 'y' or 'f(x)') that the function can produce. It's the set of all the 'y' values that result when you plug in all the valid 'x' values from the domain. Determining the range often requires a deeper understanding of the function's behavior, such as its maximum and minimum values, its increasing and decreasing intervals, and its overall shape. For example, a quadratic function, which graphs as a parabola, will have a range that is either bounded above or bounded below, depending on whether the parabola opens upwards or downwards.

To find the range, you might need to consider the function's critical points (where the derivative is zero or undefined), its end behavior (what happens as 'x' approaches positive or negative infinity), and any other key features that influence its output values. Visualizing the function's graph is often extremely helpful in determining the range, as it provides a clear picture of the set of all possible 'y' values.

Now, let's apply these concepts to our specific function, f(x) = -(x+3)(x-1). This function is a quadratic function, which means it will graph as a parabola. The factored form of the function, -(x+3)(x-1), gives us immediate insights into its roots, which are the x-intercepts of the graph. These roots occur where the function equals zero, which happens when x = -3 or x = 1. These are the points where the parabola crosses the x-axis.

To further understand the parabola's shape, we can expand the function into standard quadratic form: f(x) = -x² -2x + 3. The negative coefficient of the x² term (-1) tells us that the parabola opens downwards. This means the parabola has a maximum point, also known as the vertex, and the range of the function will be bounded above by the y-coordinate of this vertex.

To find the vertex, we can use the formula x = -b / 2a, where a and b are the coefficients of the x² and x terms, respectively. In our case, a = -1 and b = -2, so the x-coordinate of the vertex is x = -(-2) / (2 * -1) = -1. To find the y-coordinate of the vertex, we substitute this x-value back into the function: f(-1) = -(-1)² - 2(-1) + 3 = -1 + 2 + 3 = 4. Therefore, the vertex of the parabola is at the point (-1, 4).

Since the parabola opens downwards, the vertex represents the maximum point of the function. This means that the maximum value of f(x) is 4. The function can take on any y-value less than or equal to 4, but it will never exceed this value. This directly determines the range of the function.

Let's start by addressing the domain. As mentioned earlier, the domain of a function is the set of all possible input values (x-values) for which the function is defined. For polynomial functions, like our f(x) = -(x+3)(x-1), the domain is generally all real numbers.

To understand why, consider what operations are involved in the function. We have multiplication, addition, and subtraction. There are no square roots, denominators, or logarithms that could potentially restrict the input values. You can substitute any real number for 'x' in the expression -(x+3)(x-1), and you will always get a real number output. There are no values of 'x' that would lead to undefined results.

Therefore, the domain of the function f(x) = -(x+3)(x-1) is all real numbers. This can be expressed mathematically as (-∞, ∞), which means that 'x' can take on any value from negative infinity to positive infinity.

In simpler terms, there are no limitations on what numbers you can plug into the function. You can use positive numbers, negative numbers, zero, fractions, decimals – any real number will work. This makes the domain straightforward to determine for polynomial functions.

The range of a function is the set of all possible output values (y-values) that the function can produce. Determining the range of f(x) = -(x+3)(x-1) requires a slightly more in-depth analysis than determining the domain.

We've already established that this function is a quadratic function, which means its graph is a parabola. We also know that the parabola opens downwards because the coefficient of the x² term is negative (after expanding the function, we get f(x) = -x² -2x + 3). This is a crucial piece of information because it tells us that the parabola has a maximum point, and the range will be bounded above by the y-coordinate of this maximum point.

The maximum point of a parabola is called the vertex. We previously calculated the vertex of this parabola to be at the point (-1, 4). This means that the highest point on the graph of the function is at a y-value of 4. Since the parabola opens downwards, the function will never produce any y-values greater than 4.

Therefore, the range of the function f(x) = -(x+3)(x-1) is all real numbers less than or equal to 4. This can be expressed mathematically as (-∞, 4], where the square bracket indicates that 4 is included in the range.

In other words, the function can produce any y-value that is 4 or lower, but it will never produce a y-value that is higher than 4. This is a direct consequence of the parabola opening downwards and having a maximum point at y = 4.

Visualizing the graph of the function is extremely helpful in confirming this. The parabola extends downwards indefinitely, covering all y-values below 4, but it never goes above the y = 4 line.

In conclusion, for the function f(x) = -(x+3)(x-1):

• The domain is all real numbers, which can be written as (-∞, ∞).
• The range is all real numbers less than or equal to 4, which can be written as (-∞, 4].

Understanding the domain and range is essential for fully comprehending the behavior of a function. In this case, we've seen how the domain is unrestricted for this polynomial function, while the range is limited by the parabola's downward-opening shape and its vertex.

The analysis of the function f(x) = -(x+3)(x-1) provides a solid foundation for understanding quadratic functions in general. Quadratic functions are a fundamental topic in algebra and calculus, and they appear in various applications, from physics to engineering to economics. Further exploration of these functions can deepen your mathematical understanding and problem-solving skills.

One important aspect to consider is the relationship between the quadratic formula, the discriminant, and the nature of the roots. The quadratic formula is used to find the roots of a quadratic equation (where the function equals zero), and the discriminant (the part of the formula under the square root) tells us whether the roots are real and distinct, real and repeated, or complex. Understanding this relationship allows you to predict the behavior of the parabola without even graphing it.

Another key area to explore is transformations of quadratic functions. You can shift, stretch, and reflect parabolas by manipulating the coefficients in the quadratic equation. Understanding these transformations allows you to visualize how changes in the equation affect the graph and, consequently, the domain and range.

Furthermore, you can investigate the applications of quadratic functions in real-world scenarios. For example, projectile motion can be modeled using quadratic functions, allowing you to calculate the trajectory of a ball thrown in the air. Optimization problems, where you want to find the maximum or minimum value of a quantity, often involve quadratic functions as well.

By delving deeper into these topics, you can gain a more comprehensive understanding of quadratic functions and their significance in mathematics and its applications. This will not only enhance your problem-solving abilities but also provide a solid foundation for further mathematical studies.

To solidify your understanding of domain and range, it's beneficial to work through various practice problems. These problems might involve different types of functions, such as linear, quadratic, rational, and radical functions. By tackling a range of examples, you'll develop a more intuitive grasp of how to determine the domain and range in different situations.

For example, you might be given a function like g(x) = √(x - 2) and asked to find its domain and range. In this case, you would need to consider the restriction imposed by the square root. The expression inside the square root must be non-negative, so x - 2 ≥ 0, which means x ≥ 2. This determines the domain. The range would then be all non-negative real numbers, since the square root function always produces non-negative outputs.

Another example could involve a rational function, such as h(x) = 1 / (x + 3). Here, you would need to identify the value of 'x' that makes the denominator zero, as division by zero is undefined. In this case, x = -3 is excluded from the domain. The range would be all real numbers except for 0, as the function can approach 0 but never actually equal it.

By working through these types of problems, you'll learn to identify the key features that influence the domain and range of different functions. You'll also develop valuable problem-solving strategies that you can apply to more complex mathematical situations.

In addition to individual practice problems, consider working on problems that involve graphical analysis. Given the graph of a function, can you determine its domain and range? This skill is crucial for interpreting visual representations of functions and understanding their behavior. You can also use graphing tools to visualize functions and confirm your analytical solutions.

By actively engaging with practice problems, you'll not only reinforce your understanding of domain and range but also build your confidence in tackling mathematical challenges.