Need assistance with understanding the rational behind finding domain and range of functions such as 4x^2 and 3/x

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Need assistance with understanding the rational behind finding domain and range of functions such as 4x^2 and 3/x

Algebra
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domain is the set of x values for which the function exists range is the set of y values for which the function exists
y=x^2 looks like: |dw:1440652144059:dw| notice here the domain is all real numbers because the curve there exists for all real values on the x-axis notice the range there is [0,infty) because the curve just isn't there for negative y values
y=-x^2 looks like: |dw:1440652231574:dw| notice the domain is still all real numbers but the range is (-inf,0] since the graph just does not exist for positive y values

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y=1/(x-1) notice without graphing we can find the domain to be all real numbers but x=1 because we know we cannot divide by 0 and x-1=0 when x=1 see f(x)=1/(x-1) ...f(1)=1/(1-1)=1/0 <--we have 0 on bottom you cannot divide by 0 the domain you find visually and algebraically is (-inf,1) union (1,inf) and notice y=1/(x-1) this will never be 0 so the range is everything by y=0 so you can say that by saying (-inf,0) union (0,inf) You can see this visually as well: |dw:1440652384055:dw|
I think that cleared it up for me Thank You!
The graphs were a big help
visual aids do help in math i agree with that

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