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

[__SB]

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.

 

[ksdhart]

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.

 

[__SB]

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