Question

find the inverse of the function


f(x) = x^2 − 6x, x ≥ 3

Answer 1

Your given function will be the part of the parabola which lies to the right of the vertex (3, -9)

so y = x^2 - 6x
inverse is
x = y^2- 6y
y^2 - 6y = x
complete the square:
y^2 - 6y+ 9 = x+9
(y-3)^2= x+9
y - 3= ± √(x+9)
y = 3 ± √(x+9)

but we only used the part for x≥3 of the original, so
the inverse is
f^-1 (x) = 3 + √(x+9) , x ≥ -9

check:
let x = 4
f(4) = 16 - 24 = -8
f^-1 (-8) = 3 + √(-8+9)
= 3 + √1 = 4

let x = 5.67
f(5.67) = -1.8711
f^-1 (-1.8711) = 3 + √(-1.8711+9)
= 3+√7.1289
=3 + 2.67
= 5.67
I am very confident my inverse is correct!