Hey guys! Let's dive into understanding the domain and range of the function f(x) = scxsc. This might look a bit intimidating at first, but trust me, we'll break it down into manageable pieces. We'll explore what this function actually represents, then figure out where it's defined (the domain), and finally, what values it can produce (the range). So, buckle up, and let's get started!

Understanding the Function f(x) = scxsc

So, what exactly is f(x) = scxsc? Well, it's a bit of a cheeky way of writing something more familiar. Recall your trig functions! In this context, "sc" likely refers to the secant function, abbreviated as sec(x). So we can rewrite f(x) = scxsc as f(x) = sec(x) * csc(x). This function is a product of two trigonometric functions which is the secant and the cosecant. Remember that secant is the reciprocal of the cosine function, meaning sec(x) = 1/cos(x), and cosecant is the reciprocal of the sine function, meaning csc(x) = 1/sin(x). Therefore, we can rewrite our function as f(x) = (1/cos(x)) * (1/sin(x)) which simplifies to f(x) = 1 / (sin(x)cos(x)). This representation helps us to identify potential points of discontinuity. To further simplify, we can multiply both the numerator and denominator by 2, yielding f(x) = 2 / (2sin(x)cos(x)). Applying the trigonometric identity 2sin(x)cos(x) = sin(2x), we get f(x) = 2 / sin(2x). This final form is particularly useful for analyzing the function's behavior, especially when determining its domain and range. Understanding this transformation is crucial because it highlights where the function might be undefined. Specifically, we need to watch out for values of x that make the denominator, sin(2x), equal to zero. These are the values that will be excluded from the domain. Furthermore, recognizing that the function is essentially a reciprocal of a sine function helps in understanding its range. The range will be affected by the fact that sine values oscillate between -1 and 1, which means the reciprocal function will have values outside of this interval, heading towards positive and negative infinity as sin(2x) approaches zero. Therefore, having a firm grasp on trigonometric identities and reciprocal relationships is essential for analyzing and understanding the function f(x) = scxsc effectively.

Determining the Domain of f(x) = scxsc

The domain of a function is basically all the possible x-values that you can plug into the function without causing it to explode or give you an undefined result. For f(x) = scxsc = 2 / sin(2x), we need to make sure the denominator, sin(2x), isn't equal to zero. Why? Because division by zero is a big no-no in the math world; it makes the function undefined at that point. So, we need to find the values of x for which sin(2x) = 0. The sine function is zero at integer multiples of π (pi), meaning sin(nπ) = 0 where n is an integer. Therefore, we need to solve 2x = nπ for x. Dividing both sides by 2, we get x = nπ/2. This means that the function f(x) = scxsc is undefined at x = 0, ±π/2, ±π, ±3π/2, and so on. These are the values we need to exclude from the domain. In interval notation, we can express the domain as all real numbers except for these values. A concise way to represent this is: x ≠ nπ/2, where n is an integer. This notation tells us to take all real numbers, but exclude any value that can be obtained by plugging an integer into nπ/2. For instance, if n = 0, x ≠ 0; if n = 1, x ≠ π/2; if n = -1, x ≠ -π/2, and so forth. Graphically, these exclusions correspond to vertical asymptotes on the graph of the function. At each of these excluded x-values, the function approaches infinity or negative infinity. This behavior is a direct consequence of the denominator approaching zero. Thus, correctly identifying and excluding these points is vital for understanding the function's behavior and accurately representing its domain. So, in summary, the domain of f(x) = scxsc consists of all real numbers except those that make sin(2x) = 0, which are x = nπ/2, where n is any integer. This understanding is essential for further analysis of the function's properties and its graph.

Finding the Range of f(x) = scxsc

Now let's tackle the range, which refers to all the possible output values (y-values) that the function can produce. Remember that f(x) = scxsc can be written as f(x) = 2 / sin(2x). Since the sine function, sin(2x), oscillates between -1 and 1, i.e., -1 ≤ sin(2x) ≤ 1, we can analyze how this affects the range of f(x). When sin(2x) is at its maximum value of 1, f(x) = 2 / 1 = 2. When sin(2x) is at its minimum value of -1, f(x) = 2 / (-1) = -2. However, because sin(2x) can take any value between -1 and 1 (excluding 0, as we determined in the domain), the reciprocal 2 / sin(2x) can take on any value greater than or equal to 2, or less than or equal to -2. As sin(2x) approaches 0, the function f(x) approaches infinity (∞) or negative infinity (-∞). This happens because dividing by a very small number results in a very large number. Therefore, the range of f(x) = scxsc includes all real numbers greater than or equal to 2, and all real numbers less than or equal to -2. In interval notation, we can express this as (-∞, -2] ∪ [2, ∞). This means the function's output will never fall between -2 and 2. This range also makes sense when you think about the graph of the function. It consists of sections that extend from 2 upwards towards infinity, and from -2 downwards towards negative infinity, with gaps in between. Another important consideration is that because sin(2x) never actually equals 0, f(x) never actually reaches infinity or negative infinity. Instead, it gets arbitrarily close to these values, resulting in the asymptotes we discussed earlier. Understanding the behavior of the sine function and how reciprocals affect the range is crucial for determining the range of trigonometric functions like f(x) = scxsc. So, in short, the range of f(x) = scxsc is all real numbers less than or equal to -2 or greater than or equal to 2, represented as (-∞, -2] ∪ [2, ∞). That's all, folks! Now you know how to find the domain and range.