Question

The graph of $y=secx$is one-to-one on the

interval $[0,\frac{\mathrm{\pi }}{2})$and on the interval$(\frac{\mathrm{\pi }}{2},\mathrm{\pi }]$.

Short answer

The given statement: The graph of $y=secx$is one-to-one on the

interval $[0,\frac{\mathrm{\pi }}{2})$and on the interval$(\frac{\mathrm{\pi }}{2},\mathrm{\pi }]$is true.

Step-by-step solution

 Step 1. Given information

The function$y=secx$is one-to-one on the

interval$[0,\frac{\mathrm{\pi }}{2})$and on the interval$(\frac{\mathrm{\pi }}{2},\mathrm{\pi }]$.

Step 2. To find value of Sec x.

$Secx=\frac{1}{Cosx}$

Suppose $Cosx=0$then $x$would be multiple of $\frac{\mathrm{\pi }}{2}$.

That is, if $x=\frac{\mathrm{\pi }}{2}$then

$$\begin{gathered} Cosx=Cos\left(\frac{\mathrm{\pi }}{2}\right) \\ Cosx=0 \end{gathered}$$

and

$$\begin{gathered} Secx=\frac{1}{Cosx} \\ Secx=\frac{1}{Cos{\displaystyle \frac{\mathrm{\pi }}{2}}} \\ Secx=\frac{1}{0}=undefined \end{gathered}$$

Step 3. Graph of the functiony=secx.

From graph we observe that the given statement is true.