how can you find the range of these functions:
1) y=x/1+x
2) y=x^2/1+x^2
3) y=x/sqrt(1+x^2)
4) the domain of the function y=sqrt(-x+9) is (-infinity,9] i understand the 9 part but why is it -infinity shouldn't is it be +infinity?
thanks
You consider every possible value of x and the range is all the y values you can get out of it.
So for #1, x can be everything except -1 (then it would be undefined). For x --> negative inf, the function approaches 1. For x --> positive inf, the function also approaches 1. As x approaches -1 from the left, the function approaches positive inf. As x approaches -1 from the right, the function approaches negative inf. So I believe the range is all real numbers except +1.
For #2 it's x^2/1+x^2. That means we'll only have positive numbers. Right away y > or = 0. As x --> negative inf, the function approaches 1. As x --> positive inf, the function approaches 1. There are no undefined points. The denominator is always greater than the numerator so the function never hits 1 or above. The minimum is 0 at x = 0. So the range is [0, 1).
For #3 it's x/sqrt(1+x^2). Square roots can't be negative but that won't matter here. The lowest value of the square root is when x = 0 and you get y = 0 for that. As x --> negative inf we have a large negative number over the square root of a large negative number squared. At large values the +1 doesn't really do much to change it and so when you take the square root you get essentially -x/x which is -1. So it approaches -1. When you take x --> positive inf you essentially get a large number over the sqrt of a large number squared which again approaches 1. Note that it never reaches 1. The denominator is always greater than the numerator so we're working in between -1 and 1. There are no undefined points. Range is (-1, 1).
For #4 the domain of the function y=sqrt(-x+9) is (-infinity,9]. Why? Because the square root must be > or = 0 at all times. You figured out that 9 works because -9 + 9 = 0 and that's valid. The other end of it is -infinity because when negated you have +infinity + 9 which is certainly positive. Any number > 9 will give you a square root of a negative number and that doesn't have any real answer.
Hope this helps.
