I dont at all understand how to find the domain/range of a given function. I always get these questions wrong on an exam perhaps because I lack imagination? I have a simple function below but I don't understand how they got the "boundary". I don't get why outside this boundary I see nothing drawn even though the result should be negative infinity… X and Y always produce a Z which should represent some cone like graph…..Any ideas what is going on here and how they get the domain/range? Does anyone have tips on how I can find domain/range of a graph with ease? I don't see any algebraic way to find this domain range…seems to me people always figure it out based on intuition and experience.
Best Answer
"I don't see any algebraic way to find this domain range...seems to me people always figure it out based on intuition and experience."
This is completely correct. If there was a way to do so autonomously, then you wouldn't have to type ranges into graphing calculators.
But there are some good rules of thumb to follow. It used to be common to teach these before everyone had a computer in their pocket, if you look at old mathbooks you see.
The domain of a function is a combination of 2 things:
1) What domain are you given?
2) What input variables actually work? For example, you can't divide by zero or take square roots of negative values.
In your case, no restriction on x, y is given by the problem up front. But you have a square root to compute. So you know the argument to the square root must not be negative, so $y - x^2 \ge 0$, so $y \ge x^2$.
The range of a function is given also by 2 things:
1) End points, this includes the edges of the domain and infinity
2) Critical points
As you noticed, sometimes it's easier just to be able to visualize things and use intuition rather than working through all of the algebra. We can see just by looking at the problem that square roots can not be less than zero, and can go up to infinity, so that's probably the answer for the range.
But to be pedantic...
Plug $y = x^2$, the endpoints, into the equation, you get $f(x, y) = \sqrt 0$, so one endpoint of the range is zero. Since y and x can both be infinity, $f(x, y) = \sqrt(\infty)$ so the range can go up to infinity.
If it were possible for the equation to give negative values, we'd have to check critical points to see what the minimum negative value is, but a square root can't be negative so there is no need.
If the equation had been $g(x) = x^2 + 2x - 6$ then for the range we'd get "up to infinity" from checking endpoints and "down to -7" by checking the critical point at (-1, -7).