Draw the graph of `f(x)= |2x-1|+ |2x-3|`. Find the range of the function.

Explanation

Here `f(x) = |x-1|+ |2x-3|`
`" "={{:(1-x+3-2x",",, xlt1), (x-1+3-2x",",,1le x le (3)/(2)), (x-1+2x-3",",, x gt (3)/(2)):}`
`" "={{:(4-3x",",, xlt1), (2-x",",,1le x le (3)/(2)), (3x-4",",, x gt (3)/(2)):}`
Quick method :
Function changes definition at `x-1=0 and 2x-3=0 or x=1 and x=3//2`. Since the graph of the function is a straight line in each of the regions `(-oo, 1), (1, 3//2) and (3//2, oo)`, we need to check the values at some particular values of x to draw the graphs in each region.
`" "f(1)= 1 `
`" "f(3//2) = 0.5`
`" "f(0) = 4`
`" "f(2)= 1+1=2`
Thus, we plot the points ` A (0, 4), B(1, 1), C(3//2, 1//2) and D(2, 2)`.
Plotting these points and connecting with straight line, we have the following graph.

From the graph, the range of the function is `[0.5, oo)`.