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

Question

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

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Answer 1 · 2018-03-05T00:04:01

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

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))#