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!
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