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# help

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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

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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.

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

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
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Thanks!

May 16, 2022