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Chapter 2: Problem 30

Determine the range of each function. $$ y=4 \sec (x) $$

Short Answer

Expert verified
The range of \(y=4 \sec(x)\) is \( y \in ( -\infty, -4 ] \cup [ 4, \infty ) \).

Step by step solution

01

- Understand the Secant Function

The secant function, \( \sec(x) \), is the reciprocal of the cosine function: \( \sec(x) = \frac{1}{\cos(x)} \). To determine the range of \(y=4 \sec(x)\), first understand the behavior of \( \sec(x) \).
02

- Determine the Range of the Secant Function

The cosine function, \( \cos(x) \), has a range of \([-1, 1]\). The secant function is undefined where \( \cos(x) = 0 \) and its range excludes values between \(-1 \) and \(1 \). Therefore, \( \sec(x) \) has a range of \(( -\infty, -1 ] \cup [ 1, \infty ) \).
03

- Apply the Multiplicative Constant

For \(y = 4 \sec(x)\), multiply the range of \( \sec(x) \) by 4. This stretches the range by a factor of 4, resulting in: \( y \in ( -\infty, -4 ] \cup [ 4, \infty ) \).
04

- Combine Results

Thus, the range of the function \( y = 4 \sec(x) \) is: \( y \in ( -\infty, -4 ] \cup [ 4, \infty ) \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Secant Function
The secant function, denoted as sec(\(x\)), is a fundamental trigonometric function. It's defined as the reciprocal of the cosine function, which can be mathematically expressed as \(\sec(x) = \frac{1}{cos(\)x\()}\). To understand this better, remember that while the cosine function takes a value and outputs its cosine, the secant function takes a value and outputs the reciprocal of its cosine. Because cosine values range between -1 and 1, the secant function has unique properties.

The regions where the secant function is undefined are the places where the cosine function equals zero, which happens at \(x = (2n + 1) \times (\frac{\pi}{2})\), where \(n\) is an integer.
Reciprocal Trigonometric Functions
Reciprocal trigonometric functions, as the name implies, are derived from the basic trigonometric functions by taking their reciprocals. These include:
  • Cosecant (\(\csc(x)\)), which is the reciprocal of sine: \(\csc(x) = \frac{1}{sin(\)x\()}\).
  • Secant (\(\sec(x)\)), the reciprocal of cosine: \(\sec(x) = \frac{1}{cos(\)x\()}\).
  • Cotangent (\(\cot(x)\)), the reciprocal of tangent: \(\cot(x) = \frac{1}{tan(\)x\()}\).

Reciprocal functions have inverses of the y-values of their counterparts. This means they will have entirely different behaviors and ranges compared to their original functions. For example, while \(\cos(x)\) ranges from [-1, 1], \(\sec(x)\) has a range that excludes the interval (-1, 1).
Multiplicative Constant in Trigonometry
When dealing with trigonometric functions, applying a multiplicative constant stretches or compresses the graph of the function. For the function given, \(y = 4 \sec(x)\), the 4 is a multiplicative constant.

Applying this constant affects the range of the function. Originally, \(\sec(x)\) has a range of \((-\infty, -1] \cup [1, \infty)\). When multiplied by 4, this range stretches by a factor of 4, giving the new range as \((-\infty, -4] \cup [4, \infty)\). Simply put, each value in the original range is multiplied by 4, pushing values further apart or closer to the asymptotes.

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