Draw the graph of `f(x)= |x|-|2x-3|`. Find the range of the function.

Explanation

For `f(x)= |x|- |2x-3|`, the function changes definition at `x=0 and x=3//2`.
Since the graph of the function is a straight line in each region `(-oo, 0), (0, 3//2) and (3//2, oo)`, we need to check the values at some particular values of x to draw the graphs.
`" "f(-1)= 1-5=-4`
`" "f(0)=0-3=-3`
`" "f(3//2)= 1.5`
`" "f(2)= 2-1=1`
Thus, we plot the points `(-1, -4), (0, -3), (3//2, 1//2) and (2, 1)`.
Plotting these points and connecting with the straight line, we have the following graph.

From the graph, the range of the function is `(-oo, 1.5]`.