How to sketch $f(x)= |2x-1|-|1+3x|$

Question

How would you sketch $f(x)= |2x-1|-|1+3x|$? And how would you work out how to sketch it? Please help!

Answer 1

We know that

$|x| = \begin{cases} x & x\ge 0 \\ -x & x\lt 0 \end{cases}$

Hence break up your domain into different regions where you can simply write down your function without the use of the absolute values (like above) and then it should be clear what to sketch.

Answer 2

Step 1: Work out when $2x-1$ and $1+3x$ change signs.

Step 2: Using this information, write down the form of the equation in each region where both signs remain constant.

Step 3: Plot the result for each region.