Let \(f(x) = \begin{cases} k(x) &\text{if }x>3, \\ x^2-6x+12&\text{if }x\leq3. \end{cases}\). Find the function k(x) such that f is its own inverse.
So I don't know how to do this. What does it mean by f is its own inverse?? Somebody please explain the problem to me!
+23254 Let's get the inverse of y = x2 - 6x + 12
Interchange x and y: x = y2 - 6y + 12
x - 12 = y2 - 6y
Complete the square: x - 12 + 9 = y2 - 6y + 9
x - 3 = (y - 3)2
Take the square root (choose the negative): - sqrt(x - 3) = y - 3
Finish: - sqrt(x - 3) + 3 = y
Create k(x) to be the inverse: k(x) = - sqrt(x - 3) + 3
Now, if you choose a value for x (say, 2),
find f(2) (it's 4)
and then find f(4) (it's back to your original number, 2)
