# inverse function

• Jan 31st 2011, 08:56 AM
paulaa
inverse function
i wonder how i what the inverse function to f(x) = 2x - 3-| x - 3 | looks lika and how I salve it :)
• Jan 31st 2011, 08:58 AM
abhishekkgp
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.
• Jan 31st 2011, 09:11 AM
chisigma
For $\displaystyle 0 \le x \le 3$ is $\displaystyle y=3\ (x-2)$ that has as inverse $\displaystyle \displaystyle x= \frac{y}{3}+ 2$ ...

Kind regards

$\displaystyle \chi$ $\displaystyle \sigma$
• Jan 31st 2011, 09:13 AM
abhishekkgp
Quote:

Originally Posted by chisigma
For $\displaystyle 0 \le x \le 3$ is $\displaystyle f(x)=x$ that has as inverse itself...

Kind regards

$\displaystyle \chi$ $\displaystyle \sigma$

will not that be the case when x>=3??
• Jan 31st 2011, 10:44 AM
HallsofIvy
For $\displaystyle x\le 3$ (I don't see any reason to include the "$\displaystyle 0\le x$")
x- 3< 0 so |x- 3|= -(x-3). Then f(x)= 2x- 3- |x- 3|= 2x- 3+ (x- 3)= 3x- 6 so that $\displaystyle f^{-1}(x)= \frac{x+ 6}{3}= \frac{x}{3}+ 2$ as chi-sigma said.

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

Of course, neither of those answers the question "what is the inverse of ?"
• Jan 31st 2011, 02:27 PM
paulaa
so the function have tvo inverses ?