Chapter 6.5, Problem 35AYU

In Problems 17-40, 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.

$y=\sec \left(\frac{2\pi }{3}x\right)\text{+}\text{2}$

To determine

To find: Domain and range.

Answer to Problem 35AYU

Solution:

Domain is $\left\{x\text{|}x\ne \frac{3}{4}k,k\text{ is an odd integer}\right\}$.

Range $=\left\{y│y\le 1\text{ or }y\ge 3\right\}$.

Explanation of Solution

Given:

$y=\sec \left(\frac{2\pi }{3}x\right)+2$

Calculation:

The graph $y=\sec \left(\frac{2\pi }{3}x\right)+2$.

By the graph we determine that the domain of this function is the set of all real numbers where cosecant is defined, which excludes points $x$ where vertical asymptotes occur. Since $\sec \left(\frac{2\pi }{3}x\right)=\frac{1}{\cos \left(\frac{2\pi }{3}x\right)}$ this happens when $\cos \left(\frac{2\pi }{3}x\right)=0\Rightarrow x=\left(\frac{2\pi }{3}x\right)=\frac{\left(2k+1\right)\pi }{2}\Rightarrow x=\frac{3\left(2k+1\right)\pi }{4}$. Therefore the domain is $\left\{x\text{|}x\ne \frac{3\left(2k+1\right)\pi }{4},k\text{ is an integer}\right\}$. To determine the range we take note of the value of $A=1$, which will scale the underlying reciprocal sine function amplitude to 1. The given function $\sec \left(\frac{2\pi }{3}x\right)$ will not enter into the space of the cosine $y=\cos \left(\frac{2\pi }{3}x\right)$ (apart from its maxima and minima). The range $\left\{y│y\le 1\cup y\ge 3\right\}$. This agrees with the graph provided.

Therefore the,

Domain is $\left\{x\text{|}x\ne \frac{3}{4}k,k\text{ is an odd integer}\right\}$.

Range $=\left\{y│y\le 1\text{ or }y\ge 3\right\}$.