Chapter 6.5, Problem 25AYU

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.

$y=2\sec x$

To determine

To find: Domain and range.

Answer to Problem 25AYU

Solution:

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

Range: $\left\{y\text{|}y\le -2\text{ or }y\ge 2\right\}$.

Explanation of Solution

Given:

$y=2\sec x$

Calculation:

The graph $y=2\sec x$.

By the graph we determine that the domain of this function is the set of all real numbers where secant is defined, which excludes the points of $x$ where vertical asymptotes occur. Since, $2\sec x=\frac{2}{\cos x}$ this happens $\cos x=0$.

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

Therefore the,

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

Range: $\left\{y\text{|}y\le -2\text{ or }y\ge 2\right\}$.