Given f(x)=7-2(x-1)^2, x>= 1, determine f inverse (5).

Answer 1

original: y = 7 - 2(x-1)^2

inverse : x = 7 - 2(y-1)^2

2(y-1)^2 = 7 - x
(y-1)^2 = (7-x)/2

y-1 = ± √[(7-x)/2]

y = 1 ± √[(7-x)/2] but x ≥ 1 in the original, which means only one wing of the parabola.
When we sketch that wing and reflect it in the line
y = x we see that our inverse must be

y = 1 + √[(7-x)/2]

check: let x=3 in original, y = 7-8 = -1
let x = -1 in inverse , y = 1 + √4 = 3

let x=10, y = 7- -155
in inverse, let x = -155
y = 1 + √81 = 10