the range of the function y = 8cos(3x)-6 is (1) 2,14 (3) -14,2 -6π,6π (4) -2,14

Explanation:

Step1: Recall cosine - function range

The range of the cosine function $y = \cos(t)$ is $[-1,1]$. Here $t = 3x$, so $\cos(3x)\in[-1,1]$.

Step2: Find the range of $8\cos(3x)$

Multiply the inequality for $\cos(3x)$ by 8. We get $8\times(-1)\leq8\cos(3x)\leq8\times1$, which simplifies to $-8\leq8\cos(3x)\leq8$.

Step3: Find the range of $8\cos(3x)-6$

Subtract 6 from each part of the inequality $-8\leq8\cos(3x)\leq8$. So $-8 - 6\leq8\cos(3x)-6\leq8 - 6$, which gives $-14\leq8\cos(3x)-6\leq2$.

Answer:

(3) $[-14,2]$