determaine the domain of f(x), read the details plzzz i need your help?

Question

i have 2 problems and both of them is function which i am not good in so i need your help and plzzz write every step of how to do and how we get the right answer.
question nr. 1: given f(x)= 4x^2/(2+3x^2)
determine the domain of f(x).
determine the signs of f(x).
determine the asymptotes, if any.
Draw the graph for f(x).

question nr.2: given f(x)=x^2-x-2/ x-1
determine the domain of f(x).
determine the signs of f(x).
determine the asymptotes, if any.
draw the graph for f(x).

please help me and if you want to have my email so i can write my email to you guys so you can send me because i dont have the programe for math which is called ( mathcad) so i can not draw any graph. please help me i really need help.
plz write your answers step by step how to find out and how to draw the graph.
thank you

thank you so much . but to find the asymptotes my teacher says to take the lim and so on which i dont know how to do. please help me.

Answer 1

f(x)= 4x^2/(2+3x^2)
Domain is all the real numbers
The sign of f(x) is always positive since x^2 always positive
The asymptote is y = 4/3
The graph starts at the origin and goes to bothe the left and the right and approaches the the asymptote y = 4/3 as x approaches +/- infinity.
The asymptote is simply the quotient of the x^2 terms; that is, 4x^2/3x^2 = 4/3.

f(x)=x^2-x-2/ x-1
f(x) =(x-2)(x+1)/(x-1)
When x =1 the denominator is 0 so f(x) is dicontinuous at x=1.
So the domain is all real numbers except 1.
f(x) is positive at (-1,1) and (2, infinity)
f(x) is undefined at x =1 and negative at (-infinity,-1) and(1,2)
There is a vertical asymptote at x=1
The graph comes from - infinity, croosses the negative x-axis at -1 , crosses the y a-axis at y=2 andthe shoots up to infinity getting closer and closer to the vertcal asymptote.
The curve then rises from -infinity asymptotically to the vertical asymptote and then crosses th x-axis at x = 2 an then continues to rise to in finity as x --> infinity/

Answer 2

"Domain" just means the numbers for X which make sense, when you plug them into the equation.

f(x)= 4x^2/(2+3x^2)

As long as 2 + 3x^2 is not zero, this make sense, f(x) will be another number. But that is true for any number X.

So the domain is x = -infinity to +infinity.