+126 How do we find the inverse of f(x)=x*abs(x). How do we find \(f^-1(4) and f^-1(-4)\)
+118667 \(\text{Find the invers of }f(x)=x*|x|\)
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:
