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# Graphing min value function (Tomato Subjective 128)

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

Discussion: 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.

Now we will plot all the graphs together and then consider the portion which are ‘lowest’.

(all of them together)

(considering only the minimum portions)

### Comment:

There is a rigorous way of doing this problem.

• First we consider the inequality |x| – 1 < |x-1| – 1. This implies |x| < |x – 1|. Here we need to split the domain into three pieces.
• $$x \le 0$$. This is implies |x| = – x and |x-1| = -(x-1). Therefore – x < -(x-1) or 0 < 1. This is always true. Hence for all values of $$x \le 0$$, $$|x| – 1 < |x-1|-1$$
• $$0 \le x \le 1$$. This is implies |x| = x and |x-1| = -(x-1).Therefore $$x \le -(x-1)$$ or $$2x \le 1$$ or $$x \le \frac{1}{2}$$. Hence upto x = 1/2 we consider the graph of |x|-1. From x=1/2 to 1, we will consider the graph of |x-1|-1
• Finally we will take the case where x > 1
• Like this we consider each pair of expression and solve the inequalities.
• Finally graph whichever in lowest in whatever piece of the domain.

## Chatuspathi:

• What is this topic: Graphing Techniques
• What are some of the associated concept: Absolute Value Function, Domain splitting, Transformation of Graphs
• Where can learn these topics: Cheenta I.S.I. & C.M.I. course, discusses these topics in the ‘Calculus’ module.
• Book Suggestions: Play With Graphs (Arihant Publication)
November 20, 2015