• LOGIN
  • No products in the cart.

Profile Photo

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.

Screen Shot 2015-11-19 at 9.29.26 PM

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.
    Screen Shot 2015-11-19 at 9.30.31 PM
  • 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.
    Screen Shot 2015-11-19 at 9.30.43 PM
  • 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.
    Screen Shot 2015-11-19 at 9.30.51 PM

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

Screen Shot 2015-11-19 at 9.34.00 PM

(all of them together)

Screen Shot 2015-11-19 at 9.39.39 PM

(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

No comments, be the first one to comment !

Leave a Reply

Your email address will not be published. Required fields are marked *

© Cheenta 2017

Login

Register

FACEBOOKGOOGLE Create an Account
Create an Account Back to login/register
X