What is the graph of f(x) = |x + 1| + sgn(x+1)?

Answer 1

Let's look it step by step...

f(x) = x would be a diagonal line going through points in which y = x, such as (2;2), (-6;-6) and (0;0)

f(x) = x+1 would be a diagonal line going through points in which y = x+1, such as (2;3), (-6;-5) and (0;1),

meaning it would be the line from before, but moved one to the left.

f(x) = |x+1| uses the absolute (making all values positive) would be a v-shaped graph going through points similar to the ones above, just that all y-values are positive, such as (2;3), (-6;5) and (0;1)

Finally, sgn(x) would be once horizontal line on the left going through y = -1 and another on the right through y = 1 (this function is -1 for all negative numbers and 1 for all positive numbers).

Sgn(x+1) moves the the descending (negative slope) arm of the graph one down and the right one up by one.

https://www.desmos.com/calculator/j1e8fndho9

(Note that I had to synthesize the SGN function as Desmos does not understand it.)

Answer 2

|x+1| is a piecewise function. Note that x+1<0 when x<1.

So |x+1| =-(x+1) = -1 - x when x<0

Note that x+1> 0 when x > -1. So |x+1| = x+1 if > -1

Now the sgn(x) is 1 if x>0 and sgn(x) = -1 if x<0

So sgn(x+1) = 1 if x>-1,

sgn(x+1) =-1 if x<-1

sgn(x) =0 if x=0.

Now we put this all together.

f(x) = -x-1 -1 = -x-2 if x<-1

f(x) = 1 + 0 = 1 if x=0

f(x) = x+1 + 1 =x+2 if x>1