# How do you find the inverse of y = - |x-3| + 5?

Jun 22, 2017

$y = 8 - x \textcolor{w h i t e}{\text{xxx")andcolor(white)("xxx}} y = x - 2$
Note that the "inverse" is not a (single-valued) function.

#### Explanation:

Given $y = - \left\mid x - 3 \right\mid + 5$
{: ("if",x >=3,color(white)("XXX"),"if",x < 3), (rarr,y=-(x-3)+5,,rarr,y=3-x+5), (rarr,y=x-2,,rarr,y=8-x), (rarr,x=y+2,,rarr,x=8-y), ("Noting",,,,), (,x>=3,,,x<3), (,rarr y<=5,,,y<5) :}

Exchanging the $x$ and $y$ variables to give the "inverse":
{: (color(white)("xxx"),y=x+2,color(white)("XXX"),y=8-x), ("with",,x<=5,) :}

This makes sense if you consider the graph of $- \left\mid x - 3 \right\mid + 5$

Consider any horizontal line (which will provide the inverse values).
The inverse can not have Domain values in excess of $5$
and for every value $< 5$, the $2$ values are provided for the Range.