Graph the cube root of x+4 and then add 1

Comment
The first thing you should do with something like this is to feed it into a program that will graph it for you. A commonly used program on this site is Desmos.

The next thing to do is what you started to do. Fill out a table.

Step One
Get the graph of this function.
I have uploaded 2 graphs. The first one is just a general picture of what the graph looks like. I accepted the constants that Desmos used for the domain and range.

The second graph is a focus on the region around x = - 4 which seems to be quite strange. It almost looks like it's got a vertical line at x = - 4. Actually it just takes very little to move it up the y axis.

Step Two

Make up a table

Before starting to fell out the table, let's make sure you understand how to put this in your calculator. Take a value for x. You used - 4. I'll go to - 3
(-3 + 4) ^(1/3) + 1

(1)^(1/3) + 1

1    this is where you start your calculator.
^ or y^x or x^y              Your calculator should have 1 of these keys.
(
1
/ 
3
)
= 
+ 
1
=

You should get 2 

Try another one. This time I'll just give you the answer
x = - 2
y = (-2 + 4)^(1/3) + 1
y = 2.26

x      (x + 4)^(1/3) + 1
-3                2
-2                2.36
-1                2.44
0                 2.587    This is the y  intercept.

Step Three
You were asked for the domain and range
Domain: -infinity < x < plus infinity
Range: - infinity < x < plus infinity

The first one is correct