Domain are the restrictions on X, and range would be the restriction on y.
So as for domain, ask yourself "Can X take any value?" The answer for the problem you have provided is yes. You can plug in any number for x. Whether negative, zero, or positive. That's because for any value you plug in for x, y will be defined. So since x can take any value, we say domain would be R (the set of all real numbers, negative, zero, positive, including decimals such as 2.34).
Now for range, we have to see if there are restrictions on y. Clearly the maximum y value will be equal to 1. That's because the absolute value sign will turn any number you plug in for x to positive. That positive value will get multiplied by the negative in front of the absolute value and turn negative, and the result will get added by one. So
-|x| will always result in a negative value, except when x takes the value of zero. In that case -|0| = -0 = 0
so in this case, y will have the maximum value when x is 0, and y would be -|0| + 1 = 1
I hope you can see that any other value you plug in for x will give you a y value less than one.
For example if x=-2, y= -|-2|+1 = -(+2)+1 = -2+1= -1
As x gets larger than zero or smaller than zero, y will get smaller. For instance if x is 100, y= -100+1=-99
Please note that if x is -100, y value would come out the same (=-99) because absolute value of -100 would be +100, so it doesn't matter if x is negative or positive.
If you graph this equation on the calculator, you will see that y is equal to one when x=0 (max y value), and there will be two lines from either side, going all the way to negative infinity (since there is no lower bound and y can get as negative as it can!).
Hope this helps.
