Find the domain and range of the function.
-$$f ( x ) = - 9 \cot ^ { 2 } x$$
A) Domain: $x \neq \frac { \pi } { 2 } + n \pi$, for all integers $n$; range $[ 0 , \infty )$
B) Domain: $x \neq n \pi$, for all integers $n$; range $( - \infty , 0 ]$
C) Domain: $x \neq n \pi$, for all integers $n$; range $[ 0 , \infty )$
D) Domain: $x \neq \frac { \pi } { 2 } + n \pi$, for all integers $n$; range $( - \infty , \infty )$

Question

Quizplus · Accepted Answer

The cotangent function, $\cot(x)$, is undefined when $x = n\pi$, where $n$ is an integer, because these are the points where the sine function is zero, and cotangent is the reciprocal of the tangent function ($\cot(x) = \frac{1}{\tan(x)} = \frac{\cos(x)}{\sin(x)}$). Therefore, the domain excludes $x = n\pi$. Since $\cot^2(x)$ is always non-negative, $-9\cot^2(x)$ will always be non-positive, making the range $(-\infty, 0]$.

Find the Domain and Range of the Function f(x)=−9cot⁡2xf ( x ) = - 9 \cot ^ { 2 } xf(x)=−9cot2x

Find the Domain and Range of the Function $f ( x ) = - 9 \cot ^ { 2 } x$