Answer:
Option C
Step-by-step explanation:
We know that the function [tex]y = cos(x)[/tex] has as its domain all real numbers, and as a range [tex]-1\leq y\leq 1[/tex]
We know that the function [tex]cot(x) = \frac{cos(x)}{sin(x)}[/tex]
The denominator of the function can not be equal to 0. But [tex]sin(x) = 0[/tex] for all [tex]x = n\pi[/tex] where n is an integer number.
Therefore the domain of cot(x) are all real numbers except [tex]x = n\pi[/tex]
The range of cot(x) are all real numbers
Then, the domain of f(g(x)) is:
x ∈ Domain g and g(x) ∈ Domain of f
Where:
Domain of g: All reals except [tex]x = n\pi[/tex]
Domain of f: All reals.
This is:
Domain of f(g(x)):
All real numbers except [tex]n\pi[/tex]
Range of f(g(x)):
[tex]-1\leq y\leq 1[/tex]
Therefore the correct option is:
C. Domain: All real numbers x except x does not equal npi for all integers n; Range: -1 is less than or equal to and is less than or equal to 1.
