What is the range of function y=√(-2cos^2x+3cosx-1)

Hello Friend,here is the solution for your question

so the given function is 
y= √(-2cos²x+3cosx-1)
 i.e = √[-2(cos²x-3/2+1/2)]
i.e = √[-2(cosx-3/4)²-9/16+1/2]
i.e. = √[-2(cos-3/4)²-1/16]
i.e. = √[1/8-3(cosx=3/4)²](1)

Now here  in this equation is this quantity :-
(cosx=3/4)²(2)   is to it's minimum value then the whole equation 
i.e. = √[1/8-3(cosx=3/4)²] will be maximum and vice versa 

And we know that cosx-3/4 will be minimum if cosx=3/4
therefore put this in (1) we get 
(cosx=3/4)²=0    [ cosx=3/4]
hence the minimum value of the quantity (cosx=3/4)² is 0 

put this in equation (1) 
we get ,
i.e. = √[1/8-3(cosx=3/4)²]
   =√[1/8-3(0)]        [ because minimum value of of the quantity (cosx=3/4)² is 0 ]
     =√1/8
      =1/(2√2)

this is the maximum value now to find the minimum value 

since this is function of root so the value of y will always be ≥0 

hence the minimum value of the function y is 0 

Therefore, the range of function y is [0,1/(2√2)]

__Well,I have explained explained each and every step,do tell me if you don't understand any step._

The last graph.

Step-by-step explanation:

In the first graph, the darker triangle has one side that is 1/3 of the corresponding side of the other triangle, and one side that is 2/5 of the corresponding side of the other triangle.  This is not proportional for all sides, so this is not a dilation.

In the second graph, the darker triangle has one side that is 1/2 of the corresponding side of the other triangle, and one side that is 3/5 of the corresponding side of the other triangle.  This is not proportional for all sides, so this is not a dilation.

In the third graph, the darker triangle has one side that is 1/3 of the corresponding side of the other triangle, and one side that is 1/2 of the corresponding side of the other triangle.  This is not proportional for all sides, so this is not a dilation.

In the fourth graph, the darker triangle has one side that is 2/3 of the corresponding side of he other triangle, and another side that is 4/6=2/3 of the corresponding side of the other triangle.  Since these are both right triangles, this means the other side, the hypotenuse, will have the same ratio.  Since it is proportional for all sides, this is a dilation.