14\ntwo functions\ntwo functions (f(x)) and (g(x)) are given as (f(x)=x) and (g(x)=4sin(x)).\nwhich one of…

14\ntwo functions\ntwo functions (f(x)) and (g(x)) are given as (f(x)=x) and (g(x)=4sin(x)).\nwhich one of the following is true about the range of function (f(x)cdot g(x)).\na the range of the function is (-4,4).\nb the range of the function is (0,infty)).\nc the range of the function is ((-infty,infty)).\nd the range of the function is ((-infty,0).

14\ntwo functions\ntwo functions (f(x)) and (g(x)) are given as (f(x)=x) and (g(x)=4sin(x)).\nwhich one of the following is true about the range of function (f(x)cdot g(x)).\na the range of the function is (-4,4).\nb the range of the function is (0,infty)).\nc the range of the function is ((-infty,infty)).\nd the range of the function is ((-infty,0).

Answer

Explanation:

Step1: Find the product of the functions

Let (h(x)=f(x)\cdot g(x)=x\cdot4\sin(x) = 4x\sin(x)).

Step2: Analyze the behavior as (x\to\pm\infty)

As (x\to+\infty), consider the values of (\sin(x)) which oscillates between - 1 and 1. When (\sin(x) = 1), (h(x)=4x) (tends to (+\infty)) and when (\sin(x)=-1), (h(x)= - 4x) (tends to (-\infty)). Similarly, as (x\to-\infty), when (\sin(x)=1), (h(x)=4x) (tends to (-\infty)) and when (\sin(x)= - 1), (h(x)=-4x) (tends to (+\infty)).

Answer:

C. The range of the function is ((-\infty,\infty))