Q. 28 - 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.

$y=cot \left(x+\frac{\mathrm{\pi }}{4}\right)$.

Step-by-Step Solution

Answer

The graph of the function $y=cot(x+\frac{\mathrm{\pi }}{4})$ is,

The domain of this function is, $\left\{x|x\ne \frac{k\mathrm{\pi }}{2}, k is an odd integer\right\}$.

The range of this function is, $(-\infty ,\infty )$.

Step 1 Given function is,

$y=cot\left(x+\frac{\mathrm{\pi }}{4}\right)$

The graph of this function is,

Step 2 Find the domain and the range.

From the graph,

The domain of this function is the set of all the real numbers except at the point where the cotangent function is not defined.

These points are,

$$\begin{gathered} 2x=\pm k\mathrm{\pi } \\ \mathrm{x}=\pm \frac{\mathrm{k\pi }}{2} \end{gathered}$$

where $k$ is an odd integer.

Therefore, the domain of this function is all the points except where $x=\pm \frac{k\mathrm{\pi }}{2}$ that is, $\left\{x|x\ne \frac{k\mathrm{\pi }}{2},k is an odd integer\right\}$.

The range of this function is, the set of all the real numbers or $\left(-\infty ,\infty \right)$.