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Let $f$ be defined by\[f(x) = \left\{ \begin{array}{cl} 3-x & \text{ if } x \leq 3, \\ -x^3+2x^2+3x & \text{ if } x>3. \end{array} \right.\]Calculate $f^{-1}(0)+f^{-1}(6)$.

 May 13, 2022
 #1
avatar+118128 
+2

 

Consider

 

\(y=3-x   \;\;where\;\; x\le3\\ when \;\;x\le3,\;\;y\ge0\)

 

 

\(y=3-x   \;\;where\;\; x\le3\\ -x=y-3\\ x=3-y\\ \text{the inverse will be}\quad y=3-x\;\;where\;\;x\ge 0 \)

 

since I only am asked for the inverse values in this domain I do not need to concern myself with the original function when x>3 as this would give an inverse function domain of x<0

To show you,   if 

f(x)=-x^3+2x^2+3x then

f(3)=-27+18+9 = 0    for values bigger than 3 the value will be negative.   

 

 

\(f^{-1}(0)=3-0=3\\ f^{-1}(6)=3-6=-3\\ \)

Any question?  ask away :)

 

 Here is the actual graph

https://www.geogebra.org/classic/nxzepazs

 

and here is just the pic.

 

 May 13, 2022
edited by Melody  May 13, 2022
 #2
avatar+71 
+1

Thanks!

 May 16, 2022

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