Question

Given f(x)=cos c and g(x)=cot x, what are the domain and range of f(g(x))?

Answer 1

Answer:

Option C

Step-by-step explanation:

We know that the function $y = cos(x)$ has as its domain all real numbers, and as a range $-1\leq y\leq 1$

 We know that the function $cot(x) = \frac{cos(x)}{sin(x)}$

The denominator of the function can not be equal to 0. But $sin(x) = 0$ for all $x = n\pi$ where n is an integer number.

Therefore the domain of cot(x) are all real numbers except $x = n\pi$

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 $x = n\pi$

Domain of f: All reals.

This is:

Domain of f(g(x)):

All real numbers except $n\pi$

Range of f(g(x)):

 $-1\leq y\leq 1$

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.