4 mark for review the function f is defined by f(x)=sec(1/2(x - π/2)). which of the following describes the…

Explanation:

Step1: Recall secant - cosine relationship

The secant function is defined as $\sec(t)=\frac{1}{\cos(t)}$. So, for $y = f(x)=\sec\left(\frac{1}{2}(x - \frac{\pi}{2})\right)$, the function is undefined when $\cos\left(\frac{1}{2}(x - \frac{\pi}{2})\right)=0$.

Step2: Solve for the zero - points of cosine

We know that $\cos\theta = 0$ when $\theta=(2k + 1)\frac{\pi}{2}$, where $k\in\mathbb{Z}$. Let $\theta=\frac{1}{2}(x - \frac{\pi}{2})$. Then $\frac{1}{2}(x - \frac{\pi}{2})=(2k + 1)\frac{\pi}{2}$.

Step3: Solve the equation for x

Multiply both sides of the equation $\frac{1}{2}(x - \frac{\pi}{2})=(2k + 1)\frac{\pi}{2}$ by 2: $x-\frac{\pi}{2}=(2k + 1)\pi$. Then add $\frac{\pi}{2}$ to both sides: $x=(2k + 1)\pi+\frac{\pi}{2}=2k\pi+\pi+\frac{\pi}{2}=2k\pi+\frac{3\pi}{2}$, where $k\in\mathbb{Z}$.

Answer:

D. The domain is the set of all real numbers $x$, except when $x=\frac{3\pi}{2}+2\pi k$, where $k$ is any integer.