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A question about restrictions of inverse functions

  1. Mar 4, 2012 #1
    1. The problem statement, all variables and given/known data

    Hi. I found in the answears that the inverse of function [itex]f(x)=3-\sqrt{x-2}[/itex] is [itex]f^{-1}(x)=(3-x)^{2}+2[/itex] only if we restrict it to [itex]{x:x\leq3}[/itex]. I understand that the restriction is needed because the found inverse is a parabola (and thus not one-to-one function).

    My general question (1): how to know/find out algebraically (without drawing graphs) the needed restrictions? Is there a general way, or some intuition?

    My special question (for the above case) (2): can I chose the restriction [itex][x:x\geq3][/itex]?

    3. The attempt at a solution

    No attempt since I regard this as a general mathematical knowledge question.
     
    Last edited: Mar 4, 2012
  2. jcsd
  3. Mar 4, 2012 #2

    Mark44

    Staff: Mentor

    To find the domain and range of the inverse, look at the range and domain of the original function. Notice that I reversed the order.

    For your problem, the domain of f is x >= 2, and the range of f is y <= 3. The reason that y has to be <= 3 can be seen from the formula, f(x) = 3 - √(x - 2). Here, we are subtracting a positive number from 3, so the function value (y) can be no larger than 3.

    Since the domain and range of f are, respectively, x >= 2 and y <= 3, the domain and range of f-1 are, respectively, x <= 3 and y >= 2.
     
  4. Mar 4, 2012 #3
    Thanks
     
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