find the inverse function of n(x) = sqrt(2-x) -x

Question

non-linear function

Answer 1

Let y=n(x)=√(2-x)-x.

x+y=√(2-x), (x+y)²=2-x, x²+2xy+y²=2-x, x²+2xy+x+y²=2,

x²+x(2y+1)=2-y²,

x²+x(2y+1)+(2y+1)²/4=2-y²+(2y+1)²/4,

(x+(2y+1)/2)²=[8-4y²+(4y²+4y+1)]/4=(4y+9)/4,

x+(2y+1)/2=±√(4y+9)/2,

x=[(-2y-1)±√(4y+9)]/2. There appear to be two solutions for x which would lead to two possible inverses. However, both solutions may not satisfy the original equation.

n(1)=0, so let's substitute y=0 in this:

x=(-1±√9)/2=(-1±3)/2⇒x=1 or -2. But n(1)=0 so we must reject the negative square root.

x=[(-2y-1)+√(4y+9)]/2. Let x=m(y)=[(-2y-1)+√(4y+9)]/2.

Therefore m(x)=n^-1(x)=[(-2x-1)+√(4x+9)]/2. This is the inverse.