Problem 60 - Precalculus

Question

In Exercises \(57-66,\) state the domain and range of the functions. $$y=-4 \sec (3 x)$$

Step-by-Step Solution

Answer

Domain: All real numbers except \(x = \frac{(2k+1)\pi}{6}\). Range: \((-\infty, -4] \cup [4, \infty)\).

Step 1: Understand the Function

The function given is \(y = -4 \sec(3x)\). To determine the domain and range, we need to understand the properties of the secant function, \( \sec(x) = \frac{1}{\cos(x)} \). This function is defined where \(\cos(x) eq 0\).

Step 2: Determine the Domain

To find the domain, we exclude the values for which \(\cos(3x) = 0\). The cosine function is zero at odd multiples of \(\frac{\pi}{2}\): \[ 3x = \frac{(2k+1)\pi}{2} \text{ for } k \in \mathbb{Z} \] This gives \[ x = \frac{(2k+1)\pi}{6} \] Thus, the domain of \(y = -4 \sec(3x)\) is all real numbers except \( x = \frac{(2k+1)\pi}{6} \), where \( k \) is an integer.

Step 3: Determine the Range

The range of the secant function is all real numbers except between -1 and 1, inclusive: \((-\infty, -1] \cup [1, \infty)\). Here, since it is multiplied by -4, the range is scaled to \((-\infty, -4] \cup [4, \infty)\), considering the effect of changing the amplitude.

Key Concepts

Domain of a Function

When we talk about the domain of a function, we're looking at all the possible x-values that can be put into a function without causing any mathematical errors. Think of it as the range of ingredients you can use in a recipe. If something isn't in the domain, then it causes problems, like having a missing ingredient.For most functions, especially basic algebraic ones, the domain is all real numbers. However, trigonometric functions can be a bit tricky. Take the secant function, for example. Since \(\sec(x) = \frac{1}{\cos(x)}\), it's invalid wherever \(\cos(x) = 0\). This happens at specific points in the cosine cycle, like odd multiples of \(\frac{\pi}{2}\).For the function \(y = -4 \sec(3x)\), we use this same idea. We find that \(3x\) must not equal these troublesome points, leading us to \(x = \frac{(2k+1)\pi}{6}\). Thus, the domain excludes these x-values. In general, understanding the domain is all about figuring out what makes the function behave well. Always check for values that can cause division by zero or take square roots of negative numbers, especially with trigonometric inputs.

Range of a Function

The range of a function is like the flavors you can get from your recipe after cooking it. It touches on the y-values (outputs) that a function can produce. While the domain talks about the inputs allowed, the range is what you get out of the function.For the basic secant function, the range is \((-\infty, -1] \cup [1, \infty)\). This is due to the inverse relationship with the cosine function, where secant shoots up to infinity at places cosine drops to zero. This results in huge spikes at certain points. Given the function \(y = -4 \sec(3x)\), multiplying by -4 stretches out these values by four times their size, and flips them across the x-axis. Therefore, the range becomes \((-\infty, -4] \cup [4, \infty)\), making it clear that this transformation affects how far and in what direction the secant's spikes reach, but not their general shape. Understanding range is key to predicting how a function behaves over different inputs.

Trigonometric Functions

Trigonometric functions may sound complicated, but they're all about understanding angles and circles. They are essential in describing periodic behavior, which is behavior that repeats over a predictable interval. This is why they are so useful in fields such as physics and engineering.The secant function, \(\sec(x)\), along with sine, cosine, tangent, and their counterparts, describes circular motion in trigonometry. Derived from the cosine function, the secant is special because it represents the reciprocal of cosine, \(\sec(x) = \frac{1}{\cos(x)}\). Note how its properties are heavily influenced by this relationship.Because it's built on the cosine's cycle, secant inherits specific behaviors:

• Undefined when cosine equals zero.
• Has periodic spikes as cosine moves towards zero.

When you analyze trigonometric functions like secant, it's important to remember that these spikes or undefined points affect both domain and range. Trigonometric functions help solve real-world problems from predicting tides to modeling sound waves, making them incredibly powerful tools in science and engineering.