How to find the formula for the inverse of the following function?

f(x)=2x^2-x^4 defined on x in [0,1]

1 Answer
Mar 5, 2018

f^(-1)(y) = sqrt(1-sqrt(1-y))

Explanation:

Given:

f(x) = 2x^2-x^4" " on " "[0, 1]

Let y = f(x) to find:

0 = x^4-2x^2+y

color(white)(0) = x^4-2x^2+1+y-1

color(white)(0) = (x^2-1)^2+y-1

Hence:

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

Note that x^2-1 <= 0 for x in [0, 1]

So:

x^2-1 = -sqrt(1-y)

Add 1 to both sides to get:

x^2 = 1-sqrt(1-y)

Take the positive square root (since x in [0, 1]) to get:

x = sqrt(1-sqrt(1-y))

So:

f^(-1)(y) = sqrt(1-sqrt(1-y))