Question

Is the secant function even, odd, or neither? Is its graph symmetric? With respect to what?

Short answer

The secant function is an evenfunction.

It is symmetrical about y-axis.

Step-by-step solution

Step 1. Given Information

We have to check that the secant function is an even function, odd function or neither.

Also we have to check that it is symmetric or not.

If yes, then we also to find that it is symmetric about what.

Step 2. To check that Secant function is even, odd or neither.

We know that a function is even if $f(-x)=f\left(x\right)$and is odd if $f(-x)=-f\left(x\right)$.

We also know that :-

$sec\left(-\theta \right)=sec\theta$.

So we can conclude that Secant function is an even function.

Step 3. To check about symmetric

To check the symmetry of the secant function we will use graphing utility.

By using graphing utility, we can graph the secant function as following :-

From the graph we can see that the graph is neither symmetric about x-axis nor origin, but it is symmetric about y-axis.

So we can conclude that secant function is symmetric about y-axis.