It's a piecewise function so it can take on simultaneous properties such as x being less than 2 in a certain range.
The absolute value function can be decomposed into a piecewise function;
![](https://latex.codecogs.com/png.latex?\bg_white |x - 2| )
can be interpreted asand likewise
![](https://latex.codecogs.com/png.latex?\bg_white y = x-2 )
from the domain
![](https://latex.codecogs.com/png.latex?\bg_white x>2 )
as you can consider two to be the "turning point" (since its the x-intercept) [and for technicality, range is
![](https://latex.codecogs.com/png.latex?\bg_white (0,\infty) )
.
The regular graph:
View attachment 33046
Decomposed to piecewise
View attachment 33048
So that's how an abs. value function works. The question adds a regular equation "x+1". I assume that cambridge intends you to sketch these graphs and add the oordinates.
View attachment 33050
You would consider the x-intercept for the abs. value graph and the oordinate for x+1. You would get the new value when adding the graphs to be y=3. Then you pretty much add both sides, you can sort of imagine how the graph will end up like. x+1 is negative for values less than -1 and the abs. value function is increasing (y-axis) at the same rate. This is where the piecewise function helps;
So when adding the x+1 graph,
Likewise, you can picture the graph is only increasing to the right of the abs. value intercept as both x+1 and | x - 2 | is increasing at the same rate.
![](https://latex.codecogs.com/png.latex?\bg_white y = x - 2 for x>2 )
(from the piecewise)
Hence you would get the new graph;
View attachment 33051
As you can see its pretty much the combination of both functions we got above.
Sorry I kinda realised I went too overdepth for a simple thing but just understanding abs. value graphs as piecewise functions made graphing way more easier, this is sorta my thought process.