Determine The Domain And Range Of The Function M(x) = (x² - 3)/4. Express The Answer In Interval Notation.

In the world of mathematics, functions play a crucial role in modeling and understanding relationships between variables. One fundamental aspect of working with functions is determining their domain and range. The domain represents the set of all possible input values (often denoted as x) for which the function is defined, while the range represents the set of all possible output values (often denoted as y or m(x) in this case) that the function can produce. Understanding the domain and range of a function is essential for interpreting its behavior and applying it in various contexts. In this article, we will delve into the process of finding the domain and range of the function m(x) = (x² - 3)/4. This function is a quadratic function, and its properties will influence the domain and range we determine. By the end of this discussion, you will have a clear understanding of how to identify the domain and range of this specific function and similar functions. This involves analyzing the function's structure, identifying any restrictions on input values, and considering the function's behavior as inputs change. Let's begin by dissecting the function m(x) = (x² - 3)/4 and uncovering its domain and range.

Before we dive into the specifics of the function m(x) = (x² - 3)/4, let's clarify the concepts of domain and range. These terms are foundational in the study of functions, and a solid understanding of them is crucial for success in mathematics. The domain of a function is essentially the set of all possible input values (x-values) that the function can accept without causing any mathematical errors or undefined results. In simpler terms, it's the set of all x-values for which the function 'works'. For example, if a function involves a square root, the domain would exclude any x-values that result in a negative number under the square root, as the square root of a negative number is not a real number. Similarly, if a function involves division, the domain would exclude any x-values that make the denominator zero, as division by zero is undefined. The range of a function, on the other hand, is the set of all possible output values (y-values or m(x)-values in our case) that the function can produce when we plug in values from the domain. It's the set of all possible results we can get out of the function. Determining the range often involves analyzing the function's behavior, identifying any minimum or maximum values, and considering any restrictions on the output. For example, if a function represents the height of an object, the range would likely be limited to non-negative values, as height cannot be negative. In summary, the domain is about what goes into the function, while the range is about what comes out. Understanding both is key to fully grasping the behavior of a function.

Now, let's focus on the function at hand: m(x) = (x² - 3)/4. This is a quadratic function, which means it has the form f(x) = ax² + bx + c, where a, b, and c are constants. In our case, a = 1/4, b = 0, and c = -3/4. Recognizing the function as quadratic is important because quadratic functions have specific properties that help us determine their domain and range. The most important property of quadratic functions is that their graphs are parabolas. A parabola is a U-shaped curve that can open upwards or downwards. The direction in which the parabola opens depends on the sign of the coefficient a. If a is positive, the parabola opens upwards, and if a is negative, it opens downwards. In our function, a = 1/4, which is positive, so the parabola opens upwards. This means that the function has a minimum value but no maximum value. The vertex of the parabola represents the minimum or maximum point of the function. In our case, since the parabola opens upwards, the vertex represents the minimum point. To find the vertex, we can use the formula x = -b / 2a. In our function, b = 0, so the x-coordinate of the vertex is x = -0 / (2 * (1/4)) = 0. To find the y-coordinate of the vertex, we plug this x-value back into the function: m(0) = (0² - 3) / 4 = -3/4. So, the vertex of the parabola is at the point (0, -3/4). This information is crucial for determining the range of the function. Understanding the structure and properties of m(x) = (x² - 3)/4 as a quadratic function with an upward-opening parabola and a vertex at (0, -3/4) lays the groundwork for accurately identifying its domain and range.

When it comes to finding the domain of a function, we're essentially asking: what are all the possible input values (x-values) that we can plug into the function without encountering any mathematical roadblocks? For the function m(x) = (x² - 3)/4, we need to consider if there are any restrictions on the values of x that we can use. Common restrictions include division by zero, square roots of negative numbers, and logarithms of non-positive numbers. However, in this case, we have a quadratic function, which is a polynomial. Polynomial functions are well-behaved and don't have these types of restrictions. There are no denominators that could potentially be zero, no square roots that require non-negative inputs, and no logarithms to worry about. The function m(x) = (x² - 3)/4 is defined for all real numbers. This means that we can plug in any real number for x, and the function will produce a valid output. Therefore, the domain of m(x) is the set of all real numbers. In interval notation, we represent this as (-∞, ∞). This notation indicates that the domain extends infinitely in both the negative and positive directions. Understanding that polynomial functions like this one have domains that encompass all real numbers simplifies the process of domain determination, allowing us to focus on the more nuanced aspect of finding the range.

Now that we've established the domain of m(x) = (x² - 3)/4 as all real numbers, let's tackle the range. Recall that the range is the set of all possible output values (m(x)-values) that the function can produce. Since we know m(x) is a quadratic function represented by an upward-opening parabola, we can use this information to find its range. The fact that the parabola opens upwards tells us that the function has a minimum value, which occurs at the vertex of the parabola. We previously determined that the vertex of the parabola is at the point (0, -3/4). This means that the minimum value of the function is m(0) = -3/4. Since the parabola opens upwards, the function's values will increase as x moves away from 0 in either direction. There is no upper limit to the values that m(x) can take. As x gets larger and larger (either positively or negatively), x² also gets larger and larger, and consequently, (x² - 3)/4 gets larger and larger. Therefore, the range of m(x) includes all real numbers greater than or equal to -3/4. In interval notation, we represent this as [-3/4, ∞). The square bracket indicates that -3/4 is included in the range, while the parenthesis indicates that infinity is not included (since infinity is not a specific number). In summary, the range of m(x) = (x² - 3)/4 is all real numbers greater than or equal to -3/4, which is a direct consequence of the function being an upward-opening parabola with a minimum value at its vertex.

To formally state our findings, let's express the domain and range of m(x) = (x² - 3)/4 using interval notation. Interval notation is a standard way of representing sets of numbers, and it's particularly useful for describing domains and ranges. We've already discussed that the domain of m(x) is all real numbers. In interval notation, this is written as (-∞, ∞). The parentheses around (-∞) and (∞) indicate that these endpoints are not included in the interval, as infinity is not a specific number but rather a concept of unboundedness. This notation signifies that the domain extends without limit in both the negative and positive directions along the number line. The range of m(x), as we've determined, includes all real numbers greater than or equal to -3/4. In interval notation, this is written as [-3/4, ∞). The square bracket around -3/4 indicates that this value is included in the interval, as -3/4 is the minimum value of the function. The parenthesis around (∞), as before, indicates that infinity is not included. This notation signifies that the range starts at -3/4 and extends upwards without limit. Therefore, the complete answer, expressed in interval notation, is: Domain: (-∞, ∞) Range: [-3/4, ∞). Using interval notation provides a concise and clear way to communicate the domain and range of a function, and mastering this notation is an important skill in mathematics.

In this article, we have thoroughly explored the process of determining the domain and range of the function m(x) = (x² - 3)/4. We began by defining the concepts of domain and range, emphasizing their importance in understanding the behavior of functions. We then analyzed the function m(x), recognizing it as a quadratic function with an upward-opening parabola. This understanding was crucial for finding both the domain and the range. We determined that the domain of m(x) is all real numbers, expressed in interval notation as (-∞, ∞). This is because quadratic functions, being polynomials, have no restrictions on their input values. Next, we focused on the range. By identifying the vertex of the parabola at (0, -3/4) and recognizing that the parabola opens upwards, we concluded that the range includes all real numbers greater than or equal to -3/4. This was expressed in interval notation as [-3/4, ∞). The ability to determine the domain and range of a function is a fundamental skill in mathematics. It allows us to fully understand the function's behavior and its potential applications. By analyzing the function's structure, identifying any restrictions on input values, and considering its graphical representation, we can accurately determine its domain and range. The concepts and techniques discussed in this article provide a solid foundation for tackling similar problems with other functions. Understanding domain and range is not just about finding the right answer; it's about developing a deeper understanding of how functions work and how they can be used to model real-world phenomena.