how would i go about finding the domain of the function

Question

f(x)=(3x+1)/(sqrt(x^2+x-2))
and both the domain and range of
g(x)=(5x-3)/(2x+1)

Answer 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 = (x-1)(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
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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

Answer 2

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!