determine the range of the function. maximum value = minimum value = y = 3cos(x - π/2)+5 check

Explanation:

Step1: Recall cosine - function range

The range of the basic cosine function $y = \cos(u)$ is $[- 1,1]$, i.e., $-1\leqslant\cos(u)\leqslant1$. Here $u=x - \frac{\pi}{2}$.

Step2: Multiply by 3

Multiply the inequality $-1\leqslant\cos(x-\frac{\pi}{2})\leqslant1$ by 3. We get $-3\leqslant3\cos(x - \frac{\pi}{2})\leqslant3$.

Step3: Add 5

Add 5 to each part of the inequality $-3\leqslant3\cos(x - \frac{\pi}{2})\leqslant3$. So, $-3 + 5\leqslant3\cos(x-\frac{\pi}{2})+5\leqslant3 + 5$, which simplifies to $2\leqslant y\leqslant8$.

Answer:

maximum value = 8 minimum value = 2