Math-functions

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?

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  1. 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

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

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

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