Chapter 6.5, Problem 29AYU

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.

$y=4\sec \left(\frac{1}{2}x\right)$

To determine

To find: Domain and range.

Answer to Problem 29AYU

Solution:

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

Range $\left\{y\text{|}y\le -4\text{ or }y\ge 4\right\}$.

Explanation of Solution

Given:

$y=4\sec \left(\frac{1}{2}x\right)$

Calculation:

The graph $y=4\text{sec}\left(\frac{1}{2}x\right)$.

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 $4\sec \left(\frac{1}{2}x\right)=\frac{4}{\cos \left(\frac{1}{2}x\right)}$ this happens when $\cos \left(\frac{1}{2}x\right)=0\Rightarrow \left(\frac{1}{2}x\right)=\frac{\left(2k+1\right)\pi }{2}\Rightarrow x=\left(2k+1\right)\pi$. Therefore the domain is $\left\{x\text{|}x\ne \left(2k+1\right)\pi ,k\text{ is an integer}\right\}$. To determine the range we take note of the value of $A=-4$, which will scale the underlying reciprocal sine function amplitude to 4. The given function $4\sec \left(\frac{1}{2}x\right)$ will not enter into the space of the cosine $y=4\cos \left(\frac{1}{2}x\right)$ (apart from its maxima and minima). Therefore the range is $\left\{y\text{|}y\le -4\text{ or }y\ge 4\right\}$. This agrees with the graph provided.

Therefore the,

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

Range $\left\{y\text{|}y\le -4\text{ or }y\ge 4\right\}$.