What Is the Domain of f(x,y) = ∑(x/y)^n on the XY Plane?

In summary, the domain on the xy plane for the function f(x,y) = ∑(x/y)^n, where n goes from 0 to infinity, is defined for all values of x and y where the absolute value of the ratio x/y is less than 1 and y is not equal to 0. Any other combinations where the ratio is greater than 1 or y is equal to 0 are considered domain violations and should be excluded from the sketch.
  • #1
AKJ1
43
0

Homework Statement


Suppose we have f(x,y) = ∑(x/y)^n , n goes from 0 to infinity. What is the domain on the xy plane? Sketch it.

3. Attempt
I was thinking to look at the scenario if x/y is less than 1, or bigger than 1. If the ratio is less than 1, then I can use an idea from geometric series to write out an explicit form for f(x,y) in which it will be defined for x>y, or in other words, below the line y = x. What about if x/y is bigger than 1? I am getting stuck here as how to represent that. Also I am not certain if the approach is even correct.
 
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  • #2
AKJ1 said:

Homework Statement


Suppose we have f(x,y) = ∑(x/y)^n , n goes from 0 to infinity. What is the domain on the xy plane? Sketch it.

3. Attempt
I was thinking to look at the scenario if x/y is less than 1, or bigger than 1. If the ratio is less than 1, then I can use an idea from geometric series to write out an explicit form for f(x,y) in which it will be defined for x>y, or in other words, below the line y = x. What about if x/y is bigger than 1? I am getting stuck here as how to represent that. Also I am not certain if the approach is even correct.
The approach is almost correct. Both x and y can be negative, and you can use the sum of geometric series if |x/y|<1. Does the sum exist in the opposite case? (Is it finite?)
 
  • #3
ehild said:
The approach is almost correct. Both x and y can be negative, and you can use the sum of geometric series if |x/y|<1. Does the sum exist in the opposite case? (Is it finite?)

Oh youre right! I keep forgetting we look at the absolute value of the ratio.

The sum does not exist in the opposite case, therefore any such combination where the ratio is greater than 1 is a domain violation and should be excluded from the sketch?
 
  • #4
AKJ1 said:
The sum does not exist in the opposite case, therefore any such combination where the ratio is greater than 1 is a domain violation and should be excluded from the sketch?
Yes.And do not forget y=0, when the ratio does not exist.
 

Related to What Is the Domain of f(x,y) = ∑(x/y)^n on the XY Plane?

1. What is the "Sketch Domain" of a function?

The sketch domain of a function is the set of all possible input values (or independent variables) that can be used to evaluate the function. It is the range of values that can be graphed on the x and y axes.

2. How do you determine the sketch domain of a function?

To determine the sketch domain of a function, you need to look at the restrictions or limitations on the input values. These can include restrictions from the function itself (such as division by zero) or from the context of the problem the function represents (such as only positive values being applicable).

3. Can the sketch domain of a function change?

Yes, the sketch domain of a function can change depending on the restrictions placed on the input values. For example, if a function allows for all real numbers as input, the sketch domain would be the entire x-y plane. However, if a restriction is placed, such as only positive values being applicable, the sketch domain would change to only include positive values on the x-y plane.

4. How does the sketch domain affect the graph of a function?

The sketch domain directly affects the shape and range of the graph of a function. The boundaries of the sketch domain determine the limits of the graph, and any restrictions on the input values can result in breaks or discontinuities in the graph.

5. Why is it important to consider the sketch domain when graphing a function?

The sketch domain is important to consider when graphing a function because it helps to accurately represent the behavior and limitations of the function. It also allows for a better understanding of the relationships between the input and output values of the function, and can help identify any potential errors or issues with the function.

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