Alright, let's dive into the fascinating world of trigonometric functions, specifically focusing on finding the domain and range of the function f(x) = sec(x) ⋅ csc(x). I know, it might sound a bit intimidating at first, but trust me, we'll break it down into easy-to-understand parts. So, grab your favorite beverage, and let's get started!

Understanding the Function f(x) = sec(x) ⋅ csc(x)

First, let's get familiar with our function. f(x) = sec(x) ⋅ csc(x) involves two trigonometric functions: secant (sec(x)) and cosecant (csc(x)). To truly grasp what's going on, it's super helpful to remember their definitions in terms of sine and cosine, which most of us are more comfortable with. The secant function, sec(x), is defined as 1/cos(x), and the cosecant function, csc(x), is defined as 1/sin(x). Therefore, we can rewrite our function f(x) as:

f(x) = (1/cos(x)) ⋅ (1/sin(x)) = 1/(sin(x) ⋅ cos(x))

This form of the function gives us a clearer picture of what values of x might cause issues. Remember, in mathematics, we need to be cautious about dividing by zero because it leads to undefined results. Now that we have f(x) expressed in terms of sine and cosine, we can identify potential problem spots where the denominator, sin(x) ⋅ cos(x), equals zero. This is crucial for determining the domain of the function.

To find where sin(x) ⋅ cos(x) = 0, we need to determine the values of x for which either sin(x) = 0 or cos(x) = 0. We know that sin(x) = 0 when x is an integer multiple of π (i.e., x = nπ, where n is an integer). Similarly, cos(x) = 0 when x is π/2 plus an integer multiple of π (i.e., x = (n + 1/2)π, where n is an integer). These values of x must be excluded from the domain of f(x) to avoid division by zero, which would make the function undefined.

In summary, understanding that sec(x) and csc(x) are reciprocals of cos(x) and sin(x), respectively, is key to analyzing f(x). By rewriting the function in terms of sine and cosine, we can more easily identify the values of x that make the denominator zero. This understanding forms the basis for determining the domain of the function, which is the set of all possible x values for which the function is defined. Keep this in mind as we move forward to precisely define the domain and, later, the range of f(x).

Determining the Domain

The domain of a function is the set of all possible input values (x-values) for which the function is defined. As we established earlier, our function f(x) = sec(x) ⋅ csc(x) can be rewritten as f(x) = 1/(sin(x) ⋅ cos(x)). This immediately tells us that we need to avoid any values of x that make the denominator, sin(x) ⋅ cos(x), equal to zero. Let's identify those values.

We know that sin(x) = 0 when x = nπ, where n is any integer. This means that x cannot be 0, ±π, ±2π, ±3π, and so on. Similarly, cos(x) = 0 when x = (n + 1/2)π, where n is any integer. This means x cannot be ±π/2, ±3π/2, ±5π/2, and so on. These are the values that make our denominator zero, and thus, they must be excluded from the domain.

Now, let's consider the expression sin(x) ⋅ cos(x). Using the double-angle identity, we know that sin(2x) = 2sin(x) ⋅ cos(x). Therefore, sin(x) ⋅ cos(x) = (1/2)sin(2x). So, our function can be further simplified to:

f(x) = 1/((1/2)sin(2x)) = 2/sin(2x)

From this form, it's clear that we need to avoid values of x for which sin(2x) = 0. This occurs when 2x = nπ, which means x = nπ/2, where n is any integer. This combines both the conditions we found earlier: x cannot be integer multiples of π and x cannot be π/2 plus integer multiples of π. Thus, the domain of f(x) consists of all real numbers except for these values.

In interval notation, we can express the domain as follows:

Domain(f) = {x ∈ ℝ | x ≠ nπ/2, n ∈ ℤ}

This means that x can be any real number except for nπ/2, where n is any integer. So, we exclude all the values we identified earlier, ensuring that our function is always defined. Understanding the domain is crucial, as it sets the stage for analyzing the range of the function. By knowing what values x can take, we can better understand the possible output values of f(x).

In summary, the domain of f(x) = sec(x) ⋅ csc(x) is all real numbers except integer multiples of π/2. We arrived at this conclusion by recognizing that sec(x) = 1/cos(x) and csc(x) = 1/sin(x), and identifying the values of x that make the denominator zero. This understanding is fundamental to further analysis of the function, especially when we explore its range.

Finding the Range

Now that we've nailed down the domain, let's move on to the range of the function f(x) = sec(x) ⋅ csc(x). The range is the set of all possible output values (y-values) that the function can produce. Recall that we simplified the function to f(x) = 2/sin(2x). Analyzing this form will help us determine the range.

We know that the sine function, sin(2x), oscillates between -1 and 1, inclusive. That is:

-1 ≤ sin(2x) ≤ 1

Now, we need to consider what happens when we take the reciprocal of sin(2x). When we have a fraction, like 2/sin(2x), and the denominator varies between -1 and 1, the value of the fraction can become very large (positive or negative). Specifically:

• When sin(2x) approaches 0, 2/sin(2x) approaches infinity (positive or negative).
• When sin(2x) = 1, 2/sin(2x) = 2/1 = 2.
• When sin(2x) = -1, 2/sin(2x) = 2/(-1) = -2.

Thus, 2/sin(2x) can take any value greater than or equal to 2, or any value less than or equal to -2. In other words, f(x) will never take values between -2 and 2. There will always be a value outside of these numbers.

Therefore, the range of f(x) consists of all real numbers less than or equal to -2, and all real numbers greater than or equal to 2. In interval notation, we can express the range as:

Range(f) = (-∞, -2] ∪ [2, ∞)

This means that f(x) can take any value from negative infinity up to -2 (including -2), and any value from 2 up to positive infinity (including 2). It will never take any value between -2 and 2.

In summary, the range of f(x) = sec(x) ⋅ csc(x) is all real numbers y such that y ≤ -2 or y ≥ 2. We found this by recognizing that f(x) = 2/sin(2x) and understanding that sin(2x) varies between -1 and 1. As sin(2x) approaches 0, f(x) approaches infinity, and the minimum and maximum values of f(x) are -2 and 2, respectively. Understanding the range complements our understanding of the domain, giving us a complete picture of the behavior of the function.

Quick Recap

Alright, guys, let's do a quick recap to make sure we've got everything down pat. We tackled the function f(x) = sec(x) ⋅ csc(x), and here's what we found:

• Domain: The domain is all real numbers except x = nπ/2, where n is any integer. This is because sec(x) = 1/cos(x) and csc(x) = 1/sin(x), and we need to avoid values of x that make sin(x) or cos(x) equal to zero.
• Range: The range is all real numbers y such that y ≤ -2 or y ≥ 2. This is because f(x) can be simplified to 2/sin(2x), and sin(2x) varies between -1 and 1. The function f(x) will never take a value between -2 and 2.

By finding the domain and range, we gain a solid understanding of where the function is defined and what values it can produce. This information is essential for graphing the function, solving equations involving it, and further mathematical analysis. Keep practicing, and you'll become a pro in no time!

Practical Applications and Further Exploration

Understanding the domain and range of trigonometric functions like f(x) = sec(x) ⋅ csc(x) isn't just an academic exercise. It has practical applications in various fields, including physics, engineering, and computer graphics. In physics, these functions can model oscillatory motion or wave behavior. In engineering, they might be used to design and analyze electrical circuits or mechanical systems. In computer graphics, trigonometric functions are essential for creating realistic animations and simulations.

Moreover, exploring the properties of this function can lead to deeper insights into trigonometric identities and transformations. For instance, we used the double-angle identity sin(2x) = 2sin(x) ⋅ cos(x) to simplify the function and find its domain and range. Delving further into trigonometric identities can reveal other interesting relationships and simplifications.

In addition to analytical methods, graphical analysis can provide valuable insights into the behavior of f(x) = sec(x) ⋅ csc(x). Plotting the function using graphing software or online tools can visually confirm the domain and range we calculated. The graph will show vertical asymptotes at x = nπ/2, where the function is undefined, and it will illustrate that the function only takes values outside the interval (-2, 2).

Furthermore, consider investigating the function's symmetry, periodicity, and any local maxima or minima. These characteristics can offer a more complete understanding of the function's behavior and its relationship to other trigonometric functions. By combining analytical, graphical, and numerical methods, you can develop a comprehensive understanding of f(x) = sec(x) ⋅ csc(x) and its applications.

So keep exploring, keep questioning, and keep pushing the boundaries of your mathematical knowledge! Happy calculating, and thanks for joining me on this mathematical journey!