Q. 32 - Precalculus Enhanced with Graphing Utilities

Question

In Problems 24-32, graph each function. Each graph should contain at least two periods. Use the graph to determine the domain and the range of each function.

$y=5\cot \left(\frac{x}{3}-\frac{\pi }{4}\right)$

Step-by-Step Solution

Answer

The graph is 

 and the domain and range is $x=\pi \left(\pm 3k+\frac{3\pi }{4}\right)$ and all real numbers

Step 1: Given information

Given the function $y=5\cot \left(\frac{x}{3}-\frac{\pi }{4}\right)$

Step 2: Sketching the graph

The graph is 

Step 3: Calculate the domain and range

The domain is all real numbers except where it is not defined 

$$\begin{gathered} \frac{x}{3}-\frac{\pi }{4}=\pm k\pi \\ x=\pm k\pi +\frac{\pi }{4} \\ x=\pi \left(\pm 3k+\frac{3\pi }{4}\right) \end{gathered}$$

The domain is $x=\pi \left(\pm 3k+\frac{3\pi }{4}\right)$ and the range is all real numbers