Find the domain, range, amplitude, and period of y = -2 csc(-2x - π/4) - 1.

Steps to Solve

1. Identify the Function Type

The given function is $y = -2 \csc(-2x - \frac{\pi}{4}) - 1$. The cosecant function, $\csc(x)$, is the reciprocal of the sine function, which is undefined where $\sin(x) = 0$.

2. Determine the Domain

To find the domain, we must identify where the function $\csc(-2x - \frac{\pi}{4})$ is defined:

• Set $-2x - \frac{\pi}{4} = n\pi$, where $n$ is any integer, since the cosecant is undefined at those points.
• Rearranging gives $x = -\frac{n\pi + \frac{\pi}{4}}{2}$.
• Thus, the domain excludes the points $x = -\frac{n\pi + \frac{\pi}{4}}{2}$ for integers $n$.

3. Find the Range

The range of the cosecant function is $(-\infty, -1] \cup [1, \infty)$.

• The transformation by $-2$ and $-1$ modifies the range:
• The new range is $(-\infty, -3] \cup [1, \infty)$ because it stretches and shifts the original range.

4. Calculate the Amplitude

Cosecant functions do not have an amplitude in the traditional sense like sine or cosine; the amplitude applies to wave-like functions. Thus:

• The amplitude can be considered maximally as the absolute value of the coefficient before the cosecant function, $|-2| = 2$.

5. Determine the Period

The period of the cosecant function corresponds to that of the sine function:

• The period of $\csc(kx)$ is given by $\frac{2\pi}{|k|}$. Here, $k = -2$.
• Thus, the period is $\frac{2\pi}{2} = \pi$.