Question 1165500
<pre>

𝑓(𝑥) = {|𝑥| 𝑓𝑜𝑟 𝑥 < −2
     −𝑥 − 1 𝑓𝑜𝑟 − 2 ≤ 𝑥 < 2
   1/2𝑥 + 2 𝑓𝑜𝑟 𝑥 ≥ 2

On the left below is the graph of y = |x|.  On the right is the same graph
with all of it erased except the part where x < -2. And since it does
not have a value at -2, we put an open circle where x would be -2.

{{{drawing(200,4100/33,-6.6,6.6,-4.1,4.1,

graph(200,4100/33,-6.6,6.6,-4.1,4.1,abs(x)))}}}{{{drawing(200,4100/33,-6.6,6.6,-4.1,4.1,
circle(-2,2,.2),
graph(200,4100/33,-6.6,6.6,-4.1,4.1,
abs(x)*sqrt(-2-x)/sqrt(-2-x)))}}}

On the left below is the graph of y = -x-1.  On the right is the same graph
with all of it erased except the part where -2 ≤ x < 2. And since it has
a value where x = -2, we put a closed circle where x = -2.  But since
it does not have a value at 2, we put an open circle where x 
would be 2.

{{{drawing(200,4100/33,-6.6,6.6,-4.1,4.1,
graph(200,4100/33,-6.6,6.6,-4.1,4.1,-x-1))}}}{{{drawing(200,4100/33,-6.6,6.6,-4.1,4.1,
circle(-2,1,.2),circle(-2,1,.15),circle(-2,1,.1),circle(-2,1,.05),
circle(2,-3,.2),
graph(200,4100/33,-6.6,6.6,-4.1,4.1,(-x-1)*sqrt(2-abs(x))/sqrt(2-abs(x))))}}}

On the left below is the graph of y = 1/2x + 2.  On the right is the same
graph with all of it erased except the part where x ≥ 2. And since it does
have a value when x = 2, we put a closed circle where x = 2.

{{{drawing(200,4100/33,-6.6,6.6,-4.1,4.1,

graph(200,4100/33,-6.6,6.6,-4.1,4.1,.5x+2))}}}{{{drawing(200,4100/33,-6.6,6.6,-4.1,4.1,
circle(2,3,.2),circle(2,3,.15),circle(2,3,.1),circle(2,3,.05),
graph(200,4100/33,-6.6,6.6,-4.1,4.1,
(.5x+2)*sqrt(-2+x)/sqrt(-2+x)*sqrt(-2+abs(x))/sqrt(-2+abs(x))))}}}

To get the final graph, we put all three of those graphs on the
right above all together on the same set of axes:

{{{drawing(200,4100/33,-6.6,6.6,-4.1,4.1,
circle(2,3,.2),circle(2,3,.15),circle(2,3,.1),circle(2,3,.05),
graph(200,4100/33,-6.6,6.6,-4.1,4.1,
(.5x+2)*sqrt(-2+x)/sqrt(-2+x)*sqrt(-2+abs(x))/sqrt(-2+abs(x)))
circle(-2,1,.2),circle(-2,1,.15),circle(-2,1,.1),circle(-2,1,.05),
circle(2,-3,.2),
graph(200,4100/33,-6.6,6.6,-4.1,4.1,(-x-1)*sqrt(2-abs(x))/sqrt(2-abs(x))),
circle(-2,2,.22),
graph(200,4100/33,-6.6,6.6,-4.1,4.1,
abs(x)*sqrt(-2-x)/sqrt(-2-x))



)}}}

Edwin</pre>