Question

Sketch the graph for $y={\sec }^{-1}\sec x$.

Answer 1

Hint: In this question, we will first write secant function in terms of cos then understand the relation of $\cos x$ and ${\cos }^{-1}x$. We will observe the graph of $\cos x$ , and how they change signs and use it to plot a graph of $y={\sec }^{-1}\sec x$.
Complete step-by-step answer:
In given question, we have,
$y={\sec }^{-1}\sec x$
This can be written as
$\sec y=\sec x$.
We know that $\sec x={\displaystyle \frac{1}{\cos x}}$, therefore, we can write above equation as,
${\displaystyle \frac{1}{\cos y}}={\displaystyle \frac{1}{\cos x}}$
Cross multiplying this, we get,
$\cos x=\cos y$
This can be written as,
$y={\cos }^{-1}\cos x$.
So, the graph of $y={\sec }^{-1}\sec x$ is the same as the graph of $y={\cos }^{-1}\cos x$.
Now, $\cos x$ is a periodic function with period $2\pi$, which means its values repeat in the same pattern after $2\pi$ increases in $x$. That is, $\cos x=\cos (2\pi +x)$.
Since, cos x is periodic with period $2\pi$. Therefore, $y={\cos }^{-1}\cos x$ is also period with period $2\pi$.
Also, the domain here is a set of those values of $x$ for which ${\cos }^{-1}\cos x$ is defined. And, range is the set of values where ${\cos }^{-1}\cos x$ lies.
Now, for all real values of $x$, $\cos x$ lies between -1 and 1. And, between -1 and 1, the inverse function of cosine is defined. Therefore, ${\cos }^{-1}\cos x$ is defined for all real values of$x$.
We know, graph of $y=\cos x$ is:

We see that, in the interval $[-\pi ,\pi ]$ , for two different values of $x$ , we have the same value of $y$.
Also, from definition of cosine inverse, in this graph, we get,
${\cos }^{-1}y=x$
If we substitute $y=\cos x$ here, we get,
${\cos }^{-1}\cos x=x$.
Now, in graph of ${\cos }^{-1}\cos x$, we have,
 $y={\cos }^{-1}\cos x$
$\Rightarrow y=x$
But, in the interval $[-\pi ,\pi ]$ , for two different values of $x$ , we have the same value of $y$ .
Let those two different values be represented by $y_1,y_2$.
Now, as $x$ increases from $-\pi$ to 0, $\cos x$ increases from -1 to 1, and hence, ${\cos }^{-1}\cos x$ decreases from $\pi$ to 0. Therefore, here we will have, $y_1=-x$.
And as it increases from 0 to $\pi$, $\cos x$ decreases from 1 to -1, and hence, ${\cos }^{-1}\cos x$ increases from 0 to $\pi$. Therefore, here we will have, $y_2=x$ .
Also, from $-\pi$ to$\pi$, the length of interval is $2\pi$ and ${\cos }^{-1}\cos x$ periodic with period $2\pi$. Therefore, the rest of the graph will repeat the same as in interval $[-\pi ,\pi ]$.
Hence, the graph of ${\cos }^{-1}\cos x$ is given by:

Hence for the graph of $y={\sec }^{-1}\sec x$ is plotted above.

Note: While plotting the graph, keep in mind that for two different values of $x$ , ${\cos }^{-1}\cos x$ will have the same value in interval of length $2\pi$. So, looking at $y=x$, do not directly plot a graph of an infinite straight line.