Chapter 6.5, Problem 32AYU

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.

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

To determine

To find: Domain and range.

Answer to Problem 32AYU

Solution:

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

Range $\left\{y\text{|}y\le -3\text{ or }y\ge 3\right\}$.

Explanation of Solution

Given:

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

Calculation:

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

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

$\Rightarrow x=\left(2k+1\right)$. Therefore the domain is $\left\{x\text{|}x\ne \left(2k+1\right),kisaninteger\right\}$. 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 $-3\sec \left(\frac{\pi }{2}x\right)$ will not enter into the space of the cosine $y=-3\cos \left(\frac{\pi }{2}x\right)$ (apart from its maxima and minima). Therefore the range is $\left\{y\text{|}y\le -3\text{ or }y\ge 3\right\}$. This agrees with the graph provided.

Therefore the,

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

Range $\left\{y\text{|}y\le -3\text{ or }y\ge 3\right\}$.