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how to draw graph of sec inverse sec x

Here in this post ,we will check out how to draw the graph of sec inverse sec x i.e

f(x)=sec1sec(x)

As sec(2π+x)=secx

This is a periodic function with period 2π

Also x cannot take values like 2n+12π as undefined

We know by definition

f(x)=sec1sec(x)=x if x \in [0, \pi] – {\pi/2}

Lets check out the other values

if x \in (-\pi, 0) but x \ne – \pi/2

-\pi \leq x \leq 0
Taking negative
0< -x \leq \pi
Also sec ( -x) = sec x
So f(x) = sec^{-1} sec (x) = sec^{-1} sec (-x )= -x

if x \in (\pi, 2 \pi] but x \ne 3 \pi/2

\pi < x \leq 2\pi
Taking negative
-2\pi \leq -x < -\pi
Adding 2\pi
0 \leq 2 \pi -x < \pi
Also sec (2\pi -x) = sec x
So f(x) = sec^{-1} sec (x) = sec^{-1} sec (2 \pi – x)= 2 \pi -x

Therefore the graph will be


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