This is a Test of Mathematics Solution Subjective 128 (from ISI Entrance). The book, Test of Mathematics at 10+2 Level is Published by East West Press. This problem book is indispensable for the preparation of I.S.I. B.Stat and B.Math Entrance.

## Problem

** **Draw the graph (on plain paper) of f(x)= min { |x| -1, |x-1| – 1, |x-2|-1}

## Solution

** **The easiest way to solve this problem is to draw the graph of all these three pieces of functions mentioned, and pick ones which are minimum.

Graph of y = |x| is same as, y = x, where x is non negative. For negative values of x, it is the graph of y = x reflected about x axis.

Now we apply transformations to find the remaining graphs.

- Graph of y = |x|-1 can be found by
*lowering*the graph of y = |x| by 1 unit along y axis.

- Graph of y = |x-1| – 1 can be found by first shift the graph of y = |x| along positive direction of x axis by 1 unit, and then lowering it by 1 unit along y axis.

- Graph of y = |x-2| – 1 can be found by first shift the graph of y = |x| along positive direction of x axis by 2 unit, and then lowering it by 1 unit along y axis.