f(x)= 2sin(3x)+4 Let g(x)=5f(2x) Find the range of g

Question

f(x)= 2sin(3x)+4 Let g(x)=5f(2x) Find the range of g

Thanks for the help !

Answer 1

Let's first find the range of f(x).

Starting point is range of sin(x) is [-1,1]

Now, f(x) involved vertical stretch by a factor 2 and then vertical translation by 4 units upwards.

Range of f(x) is [-1*2 + 4 , 1*2 +4 ] i.e. [2, 6]

Transformation to g(x) involves vertical stretch by a factor 5. [ie. new y's are 5 times previous y values]

Range of g(x) is [2*5 , 6*5] ie.e [10,30]