Chapter 4.6, Problem 58E

To determine

To find: whether the function is even, odd or neither..

Answer to Problem 58E

  $f\left(x\right)=-f\left(-x\right)$ , the function $f\left(x\right)=\tan x$ is an odd function.

Explanation of Solution

Given information:

Given function is

  $f\left(x\right)=x^2-\sec x$ .

Calculation:

Now to plot this function in TI-83, proceed as below:

Press the button $Y=$ and type $Y_1=\tan X$

  

Press the button WINDOW and adjust the graphing window as shown

  

Press the button TRACE or GRAPH to obtain the graph:

  

The above graph is not symmetric about $y$ axis but there is reflection about axis, so function $f\left(x\right)=\tan x$ is an odd function.

Consider,

  $f\left(x\right)=\tan x$

Let $x=\frac{\pi }{2}$

Thus,

  $\begin{array}{c}f\left(\frac{\pi }{2}\right)=\tan \frac{\pi }{2}\\ =1\end{array}$

And

  $\begin{array}{c}f\left(-\frac{\pi }{2}\right)=\tan \left(-\frac{\pi }{2}\right)\\ =-1\end{array}$

Since, $f\left(x\right)=-f\left(-x\right)$ , the function $f\left(x\right)=\tan x$ is an odd function.