the function $g$ is given by $g(x)=3csc(pi(x + 2))-1$. which of the following describes the range of $g$?\na…

the function $g$ is given by $g(x)=3csc(pi(x + 2))-1$. which of the following describes the range of $g$?\na the range of $g$ is $-3,3$.\nb the range of $g$ is $(-infty,-2cup4,infty)$.\nc the range of $g$ is $(-infty,-3cup3,infty)$.\nd the range of $g$ is $(-infty,-4cup2,infty)$.

the function $g$ is given by $g(x)=3csc(pi(x + 2))-1$. which of the following describes the range of $g$?\na the range of $g$ is $-3,3$.\nb the range of $g$ is $(-infty,-2cup4,infty)$.\nc the range of $g$ is $(-infty,-3cup3,infty)$.\nd the range of $g$ is $(-infty,-4cup2,infty)$.

Answer

Answer:

B. The range of $g$ is $(-\infty,-2]\cup[4,\infty)$.

Explanation:

Step1: Recall range of $\csc$ function

The range of $y = \csc(t)$ is $(-\infty,- 1]\cup[1,\infty)$.

Step2: Consider the given function $g(x)=3\csc(\pi(x + 2))-1$

Let $t=\pi(x + 2)$. Then $y = 3\csc(t)-1$.

Step3: Find the range of $3\csc(t)$

Since the range of $\csc(t)$ is $(-\infty,-1]\cup[1,\infty)$, the range of $3\csc(t)$ is $(-\infty,-3]\cup[3,\infty)$ (multiply each value in the range of $\csc(t)$ by 3).

Step4: Find the range of $3\csc(t)-1$

Subtract 1 from each value in the range of $3\csc(t)$. For $y_1\in(-\infty,-3]$, $y_1-1\in(-\infty,-4]$. For $y_2\in[3,\infty)$, $y_2 - 1\in[2,\infty)$. So the range of $g(x)$ is $(-\infty,-4]\cup[2,\infty)$.