Chapter 6.5, Problem 28AYU

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.

$y=-4\sec x$

To determine

To find: Domain and range.

Answer to Problem 28AYU

Solution:

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

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

Explanation of Solution

Given:

$y=-4\sec x$

Calculation:

The graph $y=-4\sec x$.

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 $-4\sec x=\frac{-4}{\cos x}$ this happens when $\cos x=0\Rightarrow x=\frac{\left(2k+1\right)\pi }{2}$. Therefore the domain is $\left\{x\text{|}x\ne \frac{\left(2k+1\right)\pi }{2},k\text{ is an integer}\right\}$. 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 $-4\sec x$ will not enter into the space of the cosine $y=-4\cos x$ (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 $\left\{x\text{|}x\ne \frac{\left(2k+1\right)\pi }{2},k\text{ is an integer}\right\}$.

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