Problem 64 - Precalculus

Question

In Exercises \(57-66,\) state the domain and range of the functions. $$y=\frac{1}{4} \cot \left(2 \pi x+\frac{\pi}{3}\right)-3$$

Step-by-Step Solution

Answer

Domain: All real numbers except where \(x = n\frac{1}{2} - \frac{1}{6}, n \in \mathbb{Z}\). Range: All real numbers.

Step 1: Identify the Trigonometric Function

The function given is a cotangent function, specifically \( y = \frac{1}{4} \cot(2\pi x + \frac{\pi}{3}) - 3 \). The cotangent function is undefined where its argument is an integer multiple of \(\pi\).

Step 2: Determine Undefined Points for Cotangent

The cotangent function is undefined when the argument \(2\pi x + \frac{\pi}{3} = k\pi\) where \(k\) is an integer. Solve for \(x\):\(2\pi x + \frac{\pi}{3} = k\pi \Rightarrow 2\pi x = k\pi - \frac{\pi}{3}\Rightarrow x = \frac{k\pi - \frac{\pi}{3}}{2\pi}\Rightarrow x = \frac{k}{2} - \frac{1}{6}\). These are the undefined points which must be excluded from the domain.

Step 3: State the Domain of the Function

Exclude the undefined points from the set of all real numbers. The domain is all real numbers except \(x = n\frac{1}{2} - \frac{1}{6}\) where \(n\) is an integer.

Step 4: Determine the Range of the Function

The range of the basic \(\cot(x)\) function is all real numbers. Multiplying by \(\frac{1}{4}\) and subtracting \(3\) does not change the fact that the range is still all real numbers. Therefore, the range of \(y\) is all real numbers.

Key Concepts

Trigonometric Functions

Trigonometric functions are a fundamental part of mathematics, especially within the study of angles and periodic phenomena. These functions include sine, cosine, tangent, cotangent, secant, and cosecant. Each function relates to the angles and sides of a right-angled triangle.

Trigonometric functions are often used in various fields such as physics, engineering, and even biology due to their ability to model periodic patterns:

• Sine and cosine functions have a range between -1 and 1.
• Tangent and cotangent functions can take any real value.
• Secant and cosecant, inverses of cosine and sine respectively, have ranges outside of -1 to 1.

The periodic nature of these functions means they repeat their values in regular intervals when graphed. This characteristic is particularly useful in modeling waves and oscillations.

Understanding trigonometric functions involves knowing their behavior through their equations, as well as their graphical representations. The period and amplitude are key features, with the period being the distance over which the function’s shape repeats.

Cotangent Function

The cotangent function, represented as \(\cot(x)\), is the reciprocal of the tangent function. While tangent is defined as \(\tan(x) = \frac{\sin(x)}{\cos(x)}\), cotangent is written as \(\cot(x) = \frac{\cos(x)}{\sin(x)}\). Due to this inverse relationship, the cotangent function is undefined wherever the sine function equals zero.

In our example, the function is \(y = \frac{1}{4} \cot(2\pi x + \frac{\pi}{3}) - 3\). The transformation applied includes multiplication by \(\frac{1}{4}\) and a vertical shift downward by 3 units. Despite these transformations:

• The cotangent's undefined points occur when the expression within it, \(2\pi x + \frac{\pi}{3}\), equals an integer multiple of \pi\.
• The resultant undefined points for \(x\) are \(x = \frac{k}{2} - \frac{1}{6}\), where \(k\) is any integer.
• The domain excludes these points, otherwise, it includes all real numbers.
• The range for cotangent, due to its structure, is all real numbers, even when transformed.

Understanding these transformations and where cotangent is undefined aids in solving for its domain and range effectively.

Real Numbers

Real numbers are the set of all numbers that can be found on the number line. This includes all the fractions, decimals, and integers that exist between negative and positive infinity. Real numbers can be categorized into rational and irrational numbers:

• Rational numbers are numbers that can be expressed as a fraction of two integers, such as \(\frac{1}{2}\) or 3.
• Irrational numbers cannot be expressed as a simple fraction, for instance, \(\pi\) or \sqrt{2}\.

When analyzing functions like the cotangent, understanding real numbers is crucial. The domain of a function is typically all real numbers except where the function is undefined or where values would result in division by zero.

In the case of trigonometric functions, especially cotangent, where the function involves angles, undefined points are determined by the conditions where the function's calculation fails for real number inputs. By understanding both the rationality and irrationality of numbers, students can better grasp what constitutes the domain and range of trigonometric functions.