SOLUTION: Find the largest possible domain and largest possible range of the function 𝑔(𝑥) = 4 cos(3𝑥) − 3 sin(3𝑥). Give your answers in set/interval notations.

The domain is, OBVIOUSLY, the set of all real numbers, since this function (this expression) 
is defined over all this set.


To find the range, let's make this identical transformation


    4*cos(3x) - 3*sin(3x) = .   (1)


Next, notice that   +  =  =  = 1.


THEREFORE, there is such angle  that   = ,   = .

This    is simply the angle in QI, which satisfies this equation   = ,  or   = .


Then we can continue the equality (1) this way


      4*cos(3x) - 3*sin(3x) =  =  = 

          now apply the formula for sine of the sum of arguments   

    = .


Thus we presented the original expression as the sine function with amplitude 5 of argument  

    4*cos(3x) - 3*sin(3x) = .


It tells you that the range of  4*cos(3x) - 3*sin(3x)  is the interval from -5 to 5, or, in the interval form, [-5,5].


ANSWER.  The domain of the given function is the entire number line (-oo,oo).

         The range of the given function is the interval [-5,5].

This transformation and the logic, which I used, may seem as a focus - pocus.


But actually, it is a general transformation of the expression a*cos(x) - b*sin(x) with real coefficients "a" and "b"
into single harmonic function


    a*sin(x) - b*cos(x) =  = 

                        =  = 

                        =  = .


where   = .


It works always for any real coefficients "a" and "b" and transforms any linear combination a*cos(x) + b*sin(x) 

into single harmonic function    with the shift  =   and the amplitude  .


It is very useful classic trigonometric transformation and the identity to know and to use in different 
trigonometric problems.


So, it makes sense to learn and to memorize it.