A function, y = f(x), originally has a domain of x equal or bigger than 4 and a range of y equal or smaller than 1. Determine the new domain and range of y=-2f(-x+5)+1 after applying all transformations. (Hint: Sketch the graph&apply the transformations)

Note: I have tried many approaches to this and I am stuck. How do I solve this if I don't know what the original equation is?

If the domain of f(x) is x > 4

then the domain of f(-x+5) is
-x+5 > 4
x < 1

The range of f(x) is y <= 1
The range of -2f(x) is y >= -2
The range of -2f(x)+5 is y >= -2+5 or y >= 3
Consider f(x)=-x^2+1
-2f(x)+5 = 2x^2+3
Using f(-x+5) does not affect the range -- it just shifts/reflects horizontally

Thank you for response Steve. How did you get -2f(x) is y >= -2 ? I keep thinking it should be -1/2

never mind, ignore my last comment, it's incorrect. thank you for your help.

To determine the new domain and range of the function y = -2f(-x+5) + 1, we need to start by understanding the effects of the given transformations.

The first transformation is -x+5 inside the function f. This transformation moves the graph horizontally 5 units to the right. So, if the original domain was x >= 4, after this transformation, the new domain will be x >= 4 + 5 = 9.

The next transformation is multiplying the function f by -2. This transformation reflects the graph vertically and changes the sign of the y-coordinates. Since the original range was y <= 1, after this transformation, the new range will be y >= -1.

Lastly, we add 1 to the function. This transformation shifts the graph vertically 1 unit upward. After this transformation, the new range becomes y >= -1 + 1 = 0.

Combining all the transformations, we have the final new domain: x >= 9, and the final new range: y >= 0.

So, the new domain of the function y = -2f(-x+5) + 1 is x >= 9, and the new range is y >= 0.