Problem: Sketch, on plain paper, the regions represented, on the plane by the following:
Discussion: First we need to understand what |y| signifies. It is the absolute value of y, that is it is +y when y is positive and -y when y is negative.
Lets test with . Clearly then sin x = 1/2. This implies or .
Again let us test with . Then implying . But this is impossible as absolute value cannot be negative.
Using these observations we get a clear idea about what is happening.
- The values of x where sin (x) is positive (that is when , where k is any integer), we draw the graph of y = sin x and reflect it about x axis (as y = sin x and y = – sin x both satisfies the equation).
- The values of x where sin (x) is negative (that is when , where k is any integer), the given relation is not defined as absolute value of y cannot be negative.
For Part (ii)
Notice that |x| gives distance of a point from y axis and |y| gives distance of a point from x axis.
Let us split the problem into cases:
- x, y both non negative (first quadrant). Then |x| = x, |y| = y, implying is same as
- implies |x| = -x. Thus in second quadrant the inequality becomes or
- implies |x| = -x, |y| = -y. Thus in third quadrant the inequality becomes or
- implies |x| = x, |y| = -y. Thus in fourth quadrant the inequality becomes
Hence the final picture is:
(only the shaded zones, not the lines).
- What is this topic: Graphing of functions, inequalities
- What are some of the associated concept: Absolute value functions, inequalities
- 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)