How do we find the inverse of f(x)=x*abs(x). How do we find \(f^-1(4) and f^-1(-4)\)

geoNewbie21 Sep 17, 2021

#1**+1 **

\(\text{Find the invers of }f(x)=x*|x|\)

If x>=0 then f(x)=x^2 which is a very simple half concave up parabola in the first quadrant

if X < 0 then f(x)=-x^2 which is a very simple half concave down parabola in the third quadrant

To be honest, the first thing I did was sketch this. The inverse is the refection in the line y=x so I can already see where this is going.

But lets look at the algebra

let y= f(x) )it is just easier to work with)

If x >=0 then y is also positive

y=x^2

x=+sqrt(y)

so

f^-1(x)=+sqrt(x)

If x<0 then y is also negative

x=-sqrt(x)

so

f^-1(x)=-sqrt(|x|)

The tricky bit was how to put these together into one function

|x| / x = 1 if x>0 and |x| / x = -1 if x<0 so

\(f^{-1}(x)=\frac{|x|*\sqrt{|x|}}{x}\)

check:

Melody Sep 18, 2021