Question

How do you trace the graph of:
y= arcsinx + arctanx
y= ( |e^x + 2| / e^x - 1 ) e^x
Without the study of a function or the calculator

Answer 1

first of all, fix the domain of x. tell me what it is.

Answer 2

domain of arcsinx is [-1, 1]
domain of arctanx is (-infinity, infinity)

Answer 3

And after that, how do I reason?

Answer 4

for the first function, you may have memorized the sin and tan of certain angles
I would make a table (and use radians for the angle) rounded to one decimal place... this is going to be a rough graph.

example (first in degrees, because that is what I remember, then radians which is what you will plot
angle deg angle rads sin tan
0 0 0 0
30 0.5 0.5 1/sqr(3) = 0.6
45 0.8 .7 1
90 1.6 1

now go backwards
value asin atan sum
0 0 0 0
0.5 0.5 0.5? 1
0.7 0.7 0.6? 1.3
1.0 1.6 0.8 2.4

I had to guess with the tan.
plot the points (0,0), (0.5,1) (0.7, 1.3), (1, 2.4)

and connect the dots
for -1 to 0, use the same numbers but make them negative.