Question

find the domain and range of the following function. (enter your answe
g(x)=sin^{-1}(3x + 6)
domain

Explanation:

Step1: Recall domain of inverse - sine function

The domain of $y = \sin^{-1}(u)$ is $- 1\leq u\leq1$. Here $u = 3x + 6$.

Step2: Set up inequality for $x$

We have $-1\leq3x + 6\leq1$. First, solve $-1\leq3x + 6$:
Subtract 6 from both sides: $-1-6\leq3x$, i.e., $-7\leq3x$. Then divide by 3: $x\geq-\frac{7}{3}$.
Next, solve $3x + 6\leq1$:
Subtract 6 from both sides: $3x\leq1 - 6=-5$. Then divide by 3: $x\leq-\frac{5}{3}$.
So the domain of $g(x)$ is $[-\frac{7}{3},-\frac{5}{3}]$.

Step3: Recall range of inverse - sine function

The range of $y=\sin^{-1}(u)$ is $[-\frac{\pi}{2},\frac{\pi}{2}]$. Since $g(x)=\sin^{-1}(3x + 6)$, the range of $g(x)$ is $[-\frac{\pi}{2},\frac{\pi}{2}]$.

Answer:

domain: $[-\frac{7}{3},-\frac{5}{3}]$, range: $[-\frac{\pi}{2},\frac{\pi}{2}]$