If you have a multivariable function, what is the range?

__SB

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Hi all, I was hoping somebody could explain this to me.
If you have a multivariable function, what is the range?
So I was doing a question and had to find the stationarypoint of f(x,y) which turned out to be at (2,3). This was a minimum point.
I just stated the range as >= 3 as in school I had onlybeen taught the range is the y values you get from the x values.
Anybody know what it is (not the range specific to my example but in general)
Many thanks.
 
You're far from the only person I've encountered who has difficulty with this concept, and truthfully I think the way domain and range are taught comes up lacking. For a "standard" function, we have y = f(x). Thus, y is a function of only one variable, specifically x. The reason why the range is the set of y values is simply because we arbitrarily defined the function f(x) as being equal to y, to make it connect well with standard xy coordinate graphing. If we, instead, had said q=f(x), then the range would be set the q values. In your example, you were given a function of two variables. So, let's say z = f(x,y). No matter how many variables the function has, the range will always be the output and the domain the input. Let's make up a function:

\(\displaystyle z=\sqrt{x-5}+\sqrt{y+7}\)

The domain is the set of all possible inputs which produce real number outputs. In this case, that would be {x >= 5, y >= -7}

The range is the set of all real number outputs. In this case, that would be z >= 0.

Hopefully that clears some things up for you.
 
You're far from the only person I've encountered who has difficulty with this concept, and truthfully I think the way domain and range are taught comes up lacking. For a "standard" function, we have y = f(x). Thus, y is a function of only one variable, specifically x. The reason why the range is the set of y values is simply because we arbitrarily defined the function f(x) as being equal to y, to make it connect well with standard xy coordinate graphing. If we, instead, had said q=f(x), then the range would be set the q values. In your example, you were given a function of two variables. So, let's say z = f(x,y). No matter how many variables the function has, the range will always be the output and the domain the input. Let's make up a function:

\(\displaystyle z=\sqrt{x-5}+\sqrt{y+7}\)

The domain is the set of all possible inputs which produce real number outputs. In this case, that would be {x >= 5, y >= -7}

The range is the set of all real number outputs. In this case, that would be z >= 0.

Hopefully that clears some things up for you.
That's great thankyou, very simple explanation once you here it:)
 
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