Calculus Domain & Range?

Question

hello, could someone explain how i can find the following, Thanks.

How would i find G(x)= 3x+|x| / x ?

Answer 1

ok. the domain is all the numbers of x that G(x) is defined. usually, you come into problems with domain when you are dividing by 0 or taking the square root or log of a - number. in this case, we will have a problem with lxl/x. at x=0, we can not divide by 0. however, for all other x, G(x) is defined. thus, the domain of G(x) = (-infinity, 0) U (0, infinity) ** the "U" is read as "union with", so you are adding the two basically**
now, for range, the range are the values of G(x) that the function takes upon itself. in this case, because we have 3x, the function G(x) will take on ALL values. however, we also have lxl/x. for numbers < 0, lxl/x=-1 and for numbers >0 lxl/x=1. so, as x gets closer and closer to 0 from the left(negative) side, G(x) gets closer and closer to -1 and as x gets closer and closer to 0 from the right(positive) side, G(x) gets closer and closer to 1. so, the range of G(x)=(-infinity,-1) U (1, infinity)
** if you have a graphing calculator, you can graph this by putting y1=3x + abs(x)/x
you can find abs( by going to MATH > NUM > 1

Answer 2

Well the domain is fairly simple, right? It's just the set of all numbers that you can put in for x and you'll get something meaningful out of the function. Here your domain is all reals except for 0, since that would force us to divide by zero.

The range is a bit more difficult. The easiest way to get a grasp on range is to graph the function, but I'll try to explain it without a graph. First of all, I'm assuming that the x in the denominator is only dividing the |x|, let me know if that's not correct.

Alright, let's look at this. |x|/x is a rather interesting critter. For positive values, it's 1, and for negative values, it's -1, right? Okay, so let's split G into two pieces, the part for x<0 and the part for x>0.
For x<0, we have 3x-1. If x << 0 (that is, x is much less than zero), then we get a really big negative number, so the range should go to -infinity. And the biggest 3x-1 can get for negative x would be -1, but it never will actually be -1 (since x cannot be zero).
Okay, what about x>0? Then we have 3x+1. If x >> 0, then this is very large and positive, so the range should go to +infinity. And the smallest this number can get is 1, although it will never actually be 1.
So the range goes from -infinity to +infinity, but it never gets between -1 and 1. In set notation,
R= (-inf, -1) union (1, inf).

Answer 3

The domain is the set of all possible inputs (x) while the range is the set of all possible outputs (G(x) or y)
Here, x can be all numbers except 0, so that's the domain.......
As for your range, that's a little trickier, partly because it's not clear what your function is.
Do you mean G(x) = (3x + |x|)/x, or
G(x) = 3x + (|x|/x)?
If the latter, |x|/x is -1 for numbers less than 0, 1 for numbers greater than 0, and undefined for 0. So if you put 0 into this function, the function would have undefined output. Consider what happens as x approaches 0 from either side.
For numbers slightly less than 0, G(x) approaches -1
For numbers slightly more than 0, g(x) approaches 1
It appears that G(x) can never take on a value between -1 and 1, inclusive, but can take on all other values.
If you mean (3x + |x|)/x ... When I graph this it again appears to have a gap in its range again between -1 and 1. To prove this, try setting the function equal to these numbers, and a number in between, and see if a solution exists.