Chapter 6.5, Problem 40AYU

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.

$y=3\sec \left(\frac{1}{4}x\right)+\text{1}$

To determine

To find: Domain and range.

Answer to Problem 40AYU

Solution:

Domain is $\left\{x\text{|}x\ne 2\left(2k+1\right)\pi ,k\text{ is an integer}\right\}$.

Range $\left\{y\text{|}y\le -2\cup y\ge 4\right\}$.

Explanation of Solution

Given:

$y=3\sec \left(\frac{1}{4}x\right)+1$

Calculation:

The graph $y=3\sec \left(\frac{1}{4}x\right)+1$.

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 $3\sec \left(\frac{1}{4}x\right)=\frac{3}{\cos \left(\frac{1}{4}x\right)}$ this happens when $\text{cos}\left(\frac{1}{4}x\right)=0\Rightarrow \left(\frac{1}{4}x\right)=\frac{\left(2k+1\right)\pi }{2}$.

$\Rightarrow x=2\left(2k+1\right)\pi$

Therefore the domain is $\left\{x\text{|}x\ne 2\left(2k+1\right)\pi ,k\text{ is an integer}\right\}$.

To determine the range we take note of the value of $A=3$, which will scale the underlying reciprocal sine function amplitude to 3. The given function $3\sec \left(\frac{1}{4}x\right)$ will not enter into the space of the cosine $y=3\cos \left(\frac{1}{4}x\right)$ (apart from its maxima and minima). Therefore without the vertical shift, the range is going to be $\left\{y\text{|}y\le -3\cup y\ge 3\right\}$.

However the vertical shift applied this range likewise gets shifted, to $\left\{y\text{|}y\le -3+1\cup y\ge 3+1\right\}=\left\{y\text{|}y\le -2\cup y\ge 4\right\}$.

This agrees with the graph provided.

Therefore the,

Domain is $\left\{x\text{|}x\ne 2\left(2k+1\right)\pi ,k\text{ is an integer}\right\}$.

Range $\left\{y\text{|}y\le -2\cup y\ge 4\right\}$.