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

 Sep 17, 2021
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\(\text{Find the invers of }f(x)=x*|x|\)

 

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:

 

 Sep 18, 2021

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