Range of a two variable function

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tekzou

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Hi all, I'm registered on this website because Maths never been my strongest force but I'm looking to contribute to the forum the best I can.

Although, for now, I'm looking for help for one of my homework; I have a 2 variables rational function, and I need to find the range of it.

f(x,y) = (2y-2x)/(x^2+y^2+2)

Normally, I'm able to transform the equation to make appear the equation of a circle, of any other geometric form. But in this case, I can't get anything out of it. If the numerator was 1, it would easy to form a circle and draw the contour map...

But now that I have x's and y's at the numerator, what do I do ?

Thanks for the help :?
 
tekzou said:
Hi all, I'm registered on this website because Maths never been my strongest force but I'm looking to contribute to the forum the best I can.

Although, for now, I'm looking for help for one of my homework; I have a 2 variables rational function, and I need to find the range of it.

f(x,y) = (2y-2x)/(x^2+y^2+2)

Normally, I'm able to transform the equation to make appear the equation of a circle, of any other geometric form. But in this case, I can't get anything out of it. If the numerator was 1, it would easy to form a circle and draw the contour map...

But now that I have x's and y's at the numerator, what do I do ?

Thanks for the help :?
Click to expand...

suppose

f(x,y) = 1/c ............... a slice at some constant height (z = 1/c)

then

(x^2+y^2+2) = 2cy - 2cx

(x+c)2 + (y-c)2 = 2c2 - 2

What does that tell you?

Go to wolframalpha.com and look at the 3-D plot - very instructive!!
 
Last edited by a moderator:
Subhotosh Khan said:
suppose

f(x,y) = 1/c ............... a slice at some constant height (z = 1/c)

then

(x^2+y^2+2) = 2cy - 2cx

(x+c)2 + (y-c)2 = 2c2 - 2

What does that tell you?

Go to wolframalpha.com and look at the 3-D plot - very instructive!!
Click to expand...

A circle centered on where (+c,-c) is the center and root of 2c^2 - 2 the rayon ! :D

Wolframalpha is a very good website, thank you very much for sharing !
 
Well if R = root of 2c^2 - 2, it must be greater than 0. So,

2c^2 = 2

c^2 = 2/2 = 1

c= +/-1 because of the root.

if z = 1/c and c is greater than 0, z = {-1, 1} ! :D
 
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