The function f(x)=(x-5)^2+2 is not one-to-one. Identify a restricted domain that makes the function one-to-one, and find the inverse function.

a. restricted domain: x>or=5; f^-1(x)=5-sqrt(x-2)

b. restricted domain: x>or=5; f^-1(x)=5+sqrt(x-2)

c. restricted domain: x<or=5; f^-1(x)=5+sqrt(x+2)

d. restricted domain: x<or=-5; f^-1(x)=5+sqrt(x-2)

I really need help if someone could please explain this to me, I would greatly appreciate it. Thank you!

asked by Girly Girl
  1. the axis of symmetry is the line x=5
    So, any domain on one side of that line will be one-to-one.

    Now, for the inverse, swap variables and solve for f^-1

    x = (f-5)^2+2
    x-2 = (f-5)^2
    f-5 = ±√(x-2)
    f = 5±√(x-2)

    Now, we need one branch only, but √(x-2) is only defined for x>=2

    On the original parabola, the right branch is x>=5. So, only A or B is a candidate, and so B is the choice.

    posted by Steve

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