inverse function

**paulaa** · January 31st 2011, 08:56 AM

i wonder how i what the inverse function to f(x) = 2x - 3-| x - 3 | looks lika and how I salve it

**abhishekkgp** · January 31st 2011, 08:58 AM

consider two cases, viz x>=3 and x<3.
now check whether f(x) is bijection. if it is then inverse exists.
after that find the inverse of each "piece".
sketching the graph may also help.

**chisigma** · January 31st 2011, 09:11 AM

For $0 \le x \le 3$ is $y=3\ (x-2)$ that has as inverse $\displaystyle x= \frac{y}{3}+ 2$ ...

Kind regards

$\chi$ $\sigma$

**abhishekkgp** · January 31st 2011, 09:13 AM

Originally Posted by chisigma

For $0 \le x \le 3$ is $f(x)=x$ that has as inverse itself...

Kind regards

$\chi$ $\sigma$

will not that be the case when x>=3??

**HallsofIvy** · January 31st 2011, 10:44 AM

For $x\le 3$ (I don't see any reason to include the " $0\le x$ ")
x- 3< 0 so |x- 3|= -(x-3). Then f(x)= 2x- 3- |x- 3|= 2x- 3+ (x- 3)= 3x- 6 so that $f^{-1}(x)= \frac{x+ 6}{3}= \frac{x}{3}+ 2$ as chi-sigma said.

For $x\ge 3$ , |x- 3|= x- 3 so that
f(x)= 2x - 3-| x - 3 |= 2x- 3- (x- 3)= x which has inverse $f(x)= x$ .

Of course, neither of those answers the question "what is the inverse of ?"

**paulaa** · January 31st 2011, 02:27 PM

so the function have tvo inverses ?

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