+83 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 $f^{-1}(0)+f^{-1}(6)$.
+118689
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.
