Problem 59 - Precalculus

Question

In Exercises \(57-66,\) state the domain and range of the functions. $$y=2 \sec (5 x)$$

Step-by-Step Solution

Answer

Domain: All real numbers except \( x = \frac{\pi}{10} + \frac{n\pi}{5} \); Range: \(( -\infty, -2 ] \cup [ 2, \infty )\).

Step 1: Identify the Function

The given function is \( y = 2 \sec(5x) \). The function involves the secant function, which is defined as \( \sec(x) = \frac{1}{\cos(x)} \).

Step 2: Determine the Domain of \( \sec(5x) \)

Since \( \sec(x) = \frac{1}{\cos(x)} \), the secant function is undefined wherever \( \cos(x) = 0 \). This occurs at \( x = \frac{\pi}{2} + n\pi \), where \( n \) is an integer.

Step 3: Adjust for the Argument of Secant

Given \( \sec(5x) \), we need \( 5x eq \frac{\pi}{2} + n\pi \), leading to \( x eq \frac{\frac{\pi}{2} + n\pi}{5} \).

Step 4: Define the Domain

From the previous step, the domain of \( y = 2\sec(5x) \) is all real numbers except \( x = \frac{\pi}{10} + \frac{n\pi}{5} \), where \( n \) is any integer.

Step 5: Determine the Range

The secant function \( \sec(x) \) has a range of \(( -\infty, -1 ] \cup [ 1, \infty )\). Multiplying by 2, the range becomes \( (-\infty, -2 ] \cup [ 2, \infty ) \).

Step 6: State the Domain and Range

The domain of \( y = 2\sec(5x) \) is all real numbers except \( x = \frac{\pi}{10} + \frac{n\pi}{5} \), with \( n \) an integer. The range is \(( -\infty, -2 ] \cup [ 2, \infty )\).

Key Concepts

Secant Function

The secant function, denoted as \( \sec(x) \), is one of the basic trigonometric functions in mathematics. It is defined as the reciprocal of the cosine function, meaning that \( \sec(x) = \frac{1}{\cos(x)} \). Because it involves division by \( \cos(x) \), the secant function is undefined wherever the cosine function equals zero. An understanding of this function is crucial when dealing with its characteristics and behavior.

• Nature of Secant: The secant function is periodic, with a period of \( 2\pi \).
• Discontinuities: It has vertical asymptotes wherever \( \cos(x) = 0 \), which typically occurs at \( x = \frac{\pi}{2} + n\pi \), where \( n \) is an integer. These points are not in the domain.

The function greatly resembles a series of waves, bouncing between positive and negative infinity whenever it passes through these vertical asymptotes. This characteristic is fundamental to identifying its real number domain.

Domain and Range

When discussing the domain and range of a function, we are examining where the function is defined and the output values it can take.

• Domain: The domain of the secant function, specifically for \( y = 2 \sec(5x) \), is all real numbers except where \( 5x = \frac{\pi}{2} + n\pi \). This simplifies to: \( x eq \frac{\pi}{10} + \frac{n\pi}{5} \).
  Thus, the function is defined for all real numbers except these specific points.
• Range: Normally, the secant function has a range of \(( -\infty, -1 ] \cup [ 1, \infty )\).

Because we are multiplying the function by 2, the range gets scaled, becoming \(( -\infty, -2 ] \cup [ 2, \infty )\). This adjusted range means the function values remain in two distinct intervals, avoiding the area between -2 and 2.

Function Transformations

Function transformations involve changes to the basic graph of a function including translations, reflections, dilations, and compressions.

• Scaling: The given function, \( y = 2 \sec(5x) \), includes a vertical scaling by 2. This doubles the height of the existing secant function's graph.
• Frequency Changes: The transformation \( \sec(5x) \) modifies the frequency of oscillations. Typically, \( \sec(x) \) has a period of \( 2\pi \). However, multiplying \( x \) by 5 compresses the period to \( \frac{2\pi}{5} \). Therefore, the secant repeats more often over the same interval.

These transformations impact the appearance of the graph significantly, creating more frequent cycles with peaks reaching higher and lower compared to the standard secant function. Understanding these transformations can help establish how functions behave in various algebraic contexts.