Math
posted by brian on .
how would i go about finding the domain of the function
f(x)=(3x+1)/(sqrt(x^2+x2))
and both the domain and range of
g(x)=(5x3)/(2x+1)

Well, the domain would be all x except where the denominator is zero or the sqrt is of a negative number so we need to find where the denominator is zero and where x^2 + x  2 is negative.
let's look at that denominator function inside the radical sign:
d = x^2+x 2
It is a parabola that opens up
first find the zeros
0 = (x1)(x+2)
so it is zero at x = 2 and x =+1
those points must not be in the domain but also all the points between them are out because the d function dips below zero between 2 and +1
so our domain is 2 > x > +1

Now for the second one the denominator is zero for x = 1/2 and all real x except x = 1/2 is the domain
To find the range, sketch the function
for x << 0, g(x)> +5/2
for x >> 0, g(x)> +5/2
for x = 0, g(x) = 3
now look at where the numerator is zero
like
x = .6, g = 0
now look at points close to x = .5 where g(x) gets huge
like
x = .6, g = 30
x = .4, g = 25
that tells you what happens each side of the singularity at x = .5
We see that g(x) goes to +oo as x approaches .5 from the left and g(x) goes to oo as x approaches .5 from the right.
Therefore the range of g is from oo to +oo 
i understood the whole first question and then i understood the domain of the second question. i did get confused though when it came to finding the range because the teacher gave us the answers w/o explanations and it said that the range was all numbers except for 5/2. i'm not sure how she got that.
thanks for all your help!