The range of a function is all the real values that f(x) can take for values of x that lie in the domain.
y = -4*sin(51x + pi)
Here x can take values in (-inf., inf.) but for any value of x, sin(51x + pi) can only lie in the interval [-1 , 1]
=> y = -4*sin(51x + pi) lies in the interval [-4 , 4]
The range of y = -4*sin(51x + pi) is [ -4 , 4]
The range of the sine function is known to be [-1 , 1].
Therefore, we'll put:
-1 =< sin (51x + pi) =< 1.
Now, we'll multiply the inequality by -4 and we'll change it's direction.
4 >= -4*sin (51x + pi) >= -4
We'll replace the expression -4*sin (51x + pi) by y:
-4 =< y =< 4
The range of the given function is the closed interval [-4 , 4].