Hey guys! Let's dive into understanding the domain and range of the function f(x) = scxsc. It might look a bit complex at first, but we'll break it down step by step to make it super clear. We’ll explore what exactly this function means, and how to figure out all possible input values (the domain) and all possible output values (the range). Ready? Let's get started!

Understanding f(x) = scxsc

First, let’s understand what the function f(x) = scxsc actually represents. The notation "scx" might seem a bit unusual, but it stands for the secant of x multiplied by the cosecant of x. Remember your trig identities? Secant (sec x) is the reciprocal of cosine (cos x), meaning sec x = 1/cos x. Similarly, cosecant (csc x) is the reciprocal of sine (sin x), so csc x = 1/sin x. So, essentially, our function can be rewritten as:

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

Now, this looks a bit more manageable, doesn’t it? We can simplify it further using the double angle identity. Recall that sin(2x) = 2sin(x)cos(x). Thus, sin(x)cos(x) = sin(2x)/2. Substituting this back into our function, we get:

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

Therefore, f(x) = 2 csc(2x). This form of the function makes it much easier to analyze its domain and range. Understanding these transformations is key to accurately determining these properties. Remember, the goal is to rewrite the function into a form that exposes potential issues with input values and helps identify the boundaries of output values. Now that we've clarified what our function really is, we can move on to figuring out its domain. Keep this simplified form, f(x) = 2 csc(2x), in mind as we proceed, as it will be invaluable in our analysis.

Determining the Domain of f(x) = scxsc

Okay, so let’s figure out the domain of f(x) = scxsc, which we now know is equivalent to f(x) = 2/sin(2x). The domain is basically all the possible x-values that you can plug into the function without causing any mathematical mayhem. In this case, we need to watch out for division by zero. Specifically, we need to find the values of x that make the denominator, sin(2x), equal to zero.

So, we need to solve the equation:

sin(2x) = 0

Remember when the sine function equals zero? It happens at integer multiples of π (pi). That is, sin(θ) = 0 when θ = nπ, where n is any integer (…-2, -1, 0, 1, 2, …). Therefore, we have:

2x = nπ

Dividing both sides by 2, we get:

x = nπ/2

This means that the function is undefined whenever x is an integer multiple of π/2. Therefore, we need to exclude these values from the domain.

So, the domain of f(x) = scxsc is all real numbers except x = nπ/2, where n is an integer. In interval notation, we can express this as:

Domain: x ∈ ℝ, x ≠ nπ/2, where n is an integer

Or, more explicitly:

Domain: (-∞, …) ∪ (…,-π, -π/2) ∪ (-π/2, 0) ∪ (0, π/2) ∪ (π/2, π) ∪ (π, … , ∞)

Essentially, we're taking the entire real number line and punching holes in it at every integer multiple of π/2. That’s where our function becomes undefined because we’d be dividing by zero. Make sense? Great! Now, let's tackle the range.

Finding the Range of f(x) = scxsc

Alright, now that we've nailed the domain, let's zoom in on finding the range of f(x) = scxsc, which, as we know, is the same as f(x) = 2/sin(2x) or f(x) = 2 csc(2x). The range is all the possible y-values (output values) that the function can produce. To determine the range, we need to consider the properties of the sine and cosecant functions.

We know that the sine function, sin(θ), oscillates between -1 and 1, i.e., -1 ≤ sin(θ) ≤ 1. Therefore, sin(2x) also oscillates between -1 and 1: -1 ≤ sin(2x) ≤ 1.

Now, let's consider the cosecant function, csc(2x) = 1/sin(2x). Since sin(2x) is between -1 and 1, csc(2x) will be greater than or equal to 1 or less than or equal to -1. In other words, csc(2x) ≤ -1 or csc(2x) ≥ 1. It's like the sine function turned inside out; instead of being trapped between -1 and 1, it avoids that region entirely.

Our function is f(x) = 2 csc(2x). This means we're just multiplying the cosecant function by 2. So, the range of f(x) will be all values less than or equal to -2 or greater than or equal to 2. In mathematical terms:

Range: f(x) ≤ -2 or f(x) ≥ 2

In interval notation, we can write this as:

Range: (-∞, -2] ∪ [2, ∞)

So, the function f(x) = scxsc will never produce values between -2 and 2. It skips over that whole interval. Think of it like this: the sine function provides the foundation, oscillating between -1 and 1, then the reciprocal (cosecant) flips it to be outside -1 and 1, and finally, the multiplication by 2 stretches the range to be outside -2 and 2. Understanding this progression is key to intuitively grasping the function's behavior and its resulting range.

Graphing f(x) = scxsc

To really solidify your understanding of the domain and range, let's briefly talk about the graph of f(x) = scxsc, which we know is the same as f(x) = 2 csc(2x).

The graph will have vertical asymptotes at x = nπ/2, where n is an integer. These are the points where the function is undefined (remember the domain?). Between these asymptotes, the graph will consist of U-shaped curves that either open upwards (when csc(2x) is positive) or downwards (when csc(2x) is negative).

• The graph never crosses the x-axis because f(x) is never zero.
• The graph approaches the vertical asymptotes but never touches them.
• The minimum value of the upward-opening curves is 2, and the maximum value of the downward-opening curves is -2. This visually represents the range we found earlier: (-∞, -2] ∪ [2, ∞).

If you were to sketch this graph (or use a graphing calculator), you'd see a series of these U-shaped curves repeating along the x-axis, separated by the vertical asymptotes. This visual representation is an awesome way to double-check your work and intuitively understand the domain and range. The asymptotes highlight the excluded values from the domain, and the bounds of the curves illustrate the limitations of the range. It's a powerful combination!

Key Takeaways

Let's recap the most important things we've learned about the function f(x) = scxsc:

• Function: f(x) = scxsc = 2/sin(2x) = 2 csc(2x)
• Domain: All real numbers except x = nπ/2, where n is an integer. In interval notation: (-∞, …) ∪ (…,-π, -π/2) ∪ (-π/2, 0) ∪ (0, π/2) ∪ (π/2, π) ∪ (π, … , ∞)
• Range: f(x) ≤ -2 or f(x) ≥ 2. In interval notation: (-∞, -2] ∪ [2, ∞)

Remember these key points, and you'll be able to tackle similar problems with ease. Understanding the transformations and simplifications of trigonometric functions is paramount to finding domains and ranges accurately. Practice makes perfect, so keep exploring other trigonometric functions and their properties!

Practice Problems

Want to test your understanding? Try finding the domain and range of these functions:

1. g(x) = 3 csc(x/2)
2. h(x) = 1/(cos(3x)sin(3x))
3. k(x) = 5 sec(4x) csc(4x)

Work through them step-by-step, using the same techniques we discussed earlier. Simplify the functions, identify potential divisions by zero, and consider the ranges of the sine, cosine, secant, and cosecant functions. Good luck, and happy math-ing!

Conclusion

So, there you have it! We've successfully navigated the domain and range of the function f(x) = scxsc. By breaking down the function, understanding trigonometric identities, and carefully considering potential pitfalls, we were able to determine the valid input values and the possible output values. Keep practicing, and you'll become a domain and range master in no time! Remember, understanding the basics is the key to unlocking more complex mathematical concepts. Keep exploring, keep learning, and most importantly, keep having fun with math! You got this!