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Chapter 4: Problem 102

Find the period, asymptotes, and range for the function \(y=\cot (2 x+\pi / 2)\).

Short Answer

Expert verified
Period: \frac{\pi}{2}, Asymptotes: \x = \frac{n\pi}{2} - \frac{\pi}{4}, Range: (-\infty, \infty)

Step by step solution

01

Determine the period

The period of the cotangent function is \(\frac{\pi}{b}\) where \(b\) is the coefficient of \(x\). In this case, \(b = 2\). Therefore, the period is \[\frac{\pi}{2}\].
02

Find the asymptotes

The cotangent function cotangent will have vertical asymptotes where the angle inside the cotangent function equals \(n\pi\) for any integer \(n\). Set \(2x + \pi/2 = n\pi\). Solve for \(x\) to get the location of the asymptotes: \[2x + \frac{\pi}{2} = n\pi \Rightarrow x = \frac{n\pi - \frac{\pi}{2}}{2} = \frac{n\pi}{2} - \frac{\pi}{4}\].
03

Determine the range

The range of the cotangent function \(y = \cot(2x + \frac{\pi}{2})\) is all real numbers. Therefore, the range is \((-\backslashinfty, \backslashinfty)\).

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Key Concepts

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

Cotangent Function
The cotangent function, denoted as \(\text{cot}(x)\), is one of the basic trigonometric functions. It is defined as the reciprocal of the tangent function, which means \(\text{cot}(x) = \frac{\text{cos}(x)}{\text{sin}(x)}\). This function is undefined wherever the sine function is zero. As a result, cotangent has unique characteristics such as vertical asymptotes.
Period of a Function
In trigonometry, the period of a function refers to the length of one complete cycle of the function's graph. For the basic cotangent function \(\text{cot}(x)\), the period is \(\text{蟺}\) since it repeats every \(\text{蟺}\) units. However, when a cotangent function is modified by a coefficient, the period changes. The general formula to find the period for \(\text{cot}(bx)\) is \(\frac{\text{蟺}}{b}\). For instance, in the modified function \(\text{cot}(2x + \frac{\text{蟺}}{2})\), where \((b=2)\), this means the period is \(\frac{\text{蟺}}{2}\).
Vertical Asymptotes
Vertical asymptotes are lines where a function approaches infinity. For the cotangent function, vertical asymptotes occur where the sine function is zero because \(\text{cot}(x)\) is undefined at these points. To find the asymptotes for \(\text{cot}(2x + \frac{\text{蟺}}{2})\), we solve the equation inside the function for values that make the angle equal to \(\text{n蟺}\) (for any integer \(n\)). This gives us the equation: \(2x + \frac{\text{蟺}}{2} = n蟺\). By solving this equation for \(x\), we find: \(x = \frac{n蟺}{2} - \frac{蟺}{4}\). Therefore, the vertical asymptotes are located at these points.
Range of a Function
The range of a function is the set of all possible output values. For the cotangent function \(\text{cot}(x)\), the range is all real numbers \((-\backslashinfty, \backslashinfty)\). This is because the cotangent of any angle can take any real number value, excluding the points where the function is undefined (i.e., its vertical asymptotes). This property remains true for the modified function \(\text{cot}(2x + \frac{\text{蟺}}{2})\), so its range also includes all real numbers.

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Most popular questions from this chapter

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