Question

Graph each function. Be sure to label key points and show at least two cycles. Use the graph to determine the domain and the range of each function. $$ y=\frac{1}{2} \csc (2 x) $$

Short answer

The domain is: \( x \in (-\infty, \infty)\ -n\minusoth\in \subset: \(\frac{n\pi}{2)\). The range: \((-\infty, -\frac{1}{2})\ ,\frac{+\infty)\)

Step-by-step solution

- Understand the Function

The given function is y = \frac{1}{2} \csc (2x). Recall that the cosecant function, \(\text{csc}(x)\), is the reciprocal of the sine function, \(\text{sin}(x)\). So, we have \(y = \frac{1}{2} \cdot \frac{1}{\sin(2x)}\).

- Identify Key Points of the Base Function

The base function here is \(\csc(2x)\), which has undefined points (asymptotes) where \(\sin(2x) = 0\). This occurs at \(2x = n\pi\), where \(n\) is an integer. Solve for \(x\) to find the undefined points: \[ 2x = n\pi \implies x = \frac{n\pi}{2} \].

- Plot the Asymptotes

Plot vertical asymptotes at all points where \(x = \frac{n\pi}{2}\).

- Determine the Period

The period of \( \csc(kx)\) is \(\frac{2\pi}{k}\). Here, \(k = 2\), so the period of \( \csc(2x) \) is \( \frac{2\pi}{2} = \pi \).

- Plot the Function

Within one period \((0, \pi)\), \(\sin(2x)\) reaches maximum and minimum values, affecting \(\csc(2x)\). Hence,y will have peaks and troughs associated with altering \(2x\). Best practice is to plot underlying sine curve (compressed by factor of 2); reciprocal relation of \(\csc\). Scale graph for \(\frac{1}{2}\) multiplier.

- Label Key Points

Points of intersection through where \(\csc(2x)\) is defined: Detailed extremes or intercepts as opposed to underlying \(\sin(2x) \). For example, max where x=\(\pi/4\) equals to: \(\frac{1}{2}\) \(+ 2k\pi\). Reciprocal peak-trough equivalents at: \((\frac{\pi}{2})\)

- Determine Domain

The domain excludes values which make \(sin(2x) =0\), i.e., x = \frac{n \pi/2\) for integer n. Domain specified as: \(x \in (-\infty,\infty) \minus{n+12;}

- Determine Range

The range specifies output boundaries dependent upon asymptotic behavior; limit tending towards positive/negative dictions. Refers all real numbers gaps: simplified interval notation \((-\infty,-1/2) \union(1/2, +\infty)\).

Key concepts

cosecant function

The cosecant function, denoted as \(\text{csc}(x)\), is the reciprocal of the sine function. In other words, \(\text{csc}(x) = \frac{1}{\text{sin}(x)}\).
The cosecant function is undefined whenever the sine function is zero, because division by zero is undefined. This creates vertical asymptote lines in the graph where these undefined points occur.

• For example, when \(\text{sin}(x) = 0\), \(\text{csc}(x)\) is undefined.
• Graphically, the cosecant function shows curves going to infinity and negative infinity between each pair of vertical asymptotes.

Understanding the cosecant function is essential before moving on to more complex aspects like identifying vertical asymptotes and determining the period of the function.

vertical asymptotes

Vertical asymptotes are lines where the function tends towards infinity or negative infinity. They show the points where the function is undefined. For the given function \(y = \frac{1}{2} \text{csc}(2x)\), vertical asymptotes occur where the sine function equals zero.
To find these points:

• Set the argument of the sine function to zero: \(2x = n\frac{\text{π}}{2}\)
• Solve for \(x\):\begin{align*} x = \frac{n\ \text{π}}{2}\text{where } n \text{ is an integer}developing the periodic nature of asymptotes across the x-axis.

The function will then have asymptotes at every multiple of \(\frac{\text{π}}{2}\). This means the graph cannot touch or cross these vertical lines, as it approaches infinity or negative infinity instead.

function period

The period of a function is the length of one complete cycle of the function. For the general cosecant function \(y = \text{csc}(kx)\), the period is given by \(\frac{2\text{π}}{k}\).

• In our example, the function is \(y = \frac{1}{2} \text{csc}(2x)\), with \(k = 2\).
• Therefore, the period is \(\frac{2\text{π}}{2} = \text{π}\).

This tells us that the function repeats itself every \(\text{π}\) units along the x-axis. Within each period, the sine function completes a full cycle, and hence the cosecant function also completes one full cycle in this interval.

domain and range

The domain and range of a function describe the possible input (x values) and output (y values) of the function, respectively.
Domain: For the function \(y = \frac{1}{2} \text{csc}(2x)\), the domain is all real numbers except where the function is undefined:

• This occurs at \(x = \frac{n\ \text{π}}{2}\) for any integer \(n\).

Therefore, the domain is all \(x\) such that:
\(\text{Domain: } x ot= \frac{n\text{π}}{2}, n \text{ is an integer}\)
Range: The range of the cosecant function excludes the interval between -1/2 and 1/2 because the cosecant function never takes these values:

• This means the range is \((-\text{infinity}, -1/2) \cup (1/2, \text{infinity})\).

In conclusion, the range consists of all real numbers greater than or equal to 1/2 or less than or equal to -1/2.