Question

Given the graph of $y=2\sec (x+\frac{\pi }{4}).$ Select the correct statements.

• A. Range : $R-(-2,2)$
• B. Period: $\pi$
• C. Domain: $R-(4n+1)\pi /4;n\in Z$
• D. Period: $2\pi$

Answer 1

Answer: (A), (C), (D)

Period: $2\pi$

Given the graph of $y=2\sec (x+\frac{\pi }{4})$


Now, the function is not defined for $x=\frac{-7\pi }{4},\frac{-3\pi }{4},\frac{\pi }{4},\frac{5\pi }{4},\frac{9\pi }{4},\frac{13\pi }{4},...$
Or domain of the function $D:R-(4n+1)\frac{\pi }{4};n\in Z$

Also, from the graph the function ranges from $(-\infty ,-2]\cup [2,\infty )$
Thus, the range of the function: $R-(-2,2)$

Also, we know if the period of a function $f\left(x\right)=T.$
Then, the period of a function $af(bx+c)=T/|b|$
Using the same method, we know the period of $\sec x$ is $2\pi .$
Hence, the period of $2\sec (x+\frac{\pi }{4})$ is $2\pi /1$ or $2\pi$