Chapter 4.6, Problem 57E

To determine

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

Answer to Problem 57E

The function $f\left(x\right)=\sec x$ is even.

Explanation of Solution

Given information:

Given function is

  $f\left(x\right)=\sec x$ .

Calculation:

Let us consider the points to find whether the function $f\left(x\right)=\sec x$ is even, odd or neither.

Let us, first plot the graph for $f\left(x\right)=\sec x$ .

  

From the above graph, we can see that the curve of the function $f\left(x\right)=\sec x$ is symmetrical about the $y$ -axis.

Thus, the function is even.

Now we will verify algebraically whether the function $f\left(x\right)=\sec x$ is even, odd or neither.

We know, the function is even if $f\left(-x\right)\ne -f\left(x\right)$

Now let us take $x=\pi$

Substituting $x=\pi$ in $f\left(-x\right)$

We get,

  $\begin{array}{c}f\left(-x\right)=\sec \left(-x\right)\\ f\left(-\pi \right)=\sec \left(-\pi \right)\\ =-1\end{array}$

Also, substituting $x=\pi$ in $-f\left(x\right)$

We get,

  $\begin{array}{c}-f\left(x\right)=-\sec \left(x\right)\\ -f\left(\pi \right)=-\sec \left(\pi \right)\\ =-\left(-1\right)\\ =1\end{array}$

As, $f\left(-x\right)\ne -f\left(x\right)$

Hence, the function $f\left(x\right)=\sec x$ is even.