Chapter 10.1, Problem 2E

Solution

To compare: the domains and ranges of the given functions.

Given:

The given functions are:

  $\begin{array}{l}y=a\sin bx\\ y=a\cos bx\\ y=a\tan bx\end{array}$

Interpretation:

For the function $y=a\sin bx,$ the domain is all real numbers because for all real numbers, the function $\sin bx$ is defined. And, for the same function, the range is how high the sine wave goes. So, the range for the function is $-a\text{ to }a$ .

For the function $y=a\cos bx,$ the domain is all real numbers because for all real number, the function $\cos bx$ is defined. And, for the same function, the range is how high the cosine wave goes. So, the range for the function is $-a\text{ to }a$ .

Now, for the function $y=a\tan bx,$ the domain is all real numbers except when the quantity $bx$ is an odd multiple of $\frac{\pi }{2}.$ At these $x$ -values, the graph has vertical asymptotes. Hence, the domain for the function is:

  $D=R-\left\{\left.\frac{n\pi }{2b}\right\}\right.$

where $n$ is an odd integer.

And, for the same function, the range is how high the tangent trigonometry function wave goes. Hence, the range is all numbers.