Finding the range of this function of 2 variables:

In summary, the domain of the function f(x,y) = ln(9 - x2 - y2) is (x, y) in R2 such that x2 + y2 < 9. The range can be found by using the natural domain of 9 - x2 - y2 > 0, which translates to 0 ≤ x2 + y2 < 9. This is because the natural log is only defined for positive numbers, and the smallest value of x2 + y2 is 0. This explains why the range is between 0 and 9.
  • #1
JC3187
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0
f(x,y) = ln(9 - x2 - y2)

Domain I got:

(x, y) in R2 such that x2 + y2 < 9

How do I find the range from the domain?

Wouldn't it be all positive numbers of ln such that it is less than ln(9)?

Answer says less than or equal to 9, I don't get why.

Any input is appreciated!
 
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  • #2
Write the domain a little bit more precisely, and you will see why.
 
  • #3
Natural domain: 9 - x2 - y2 > 0
x2 + y2 < 9

How more precise can I be with this...
 
  • #4
0 <= x² + y² < 9
 
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  • #5
Still am confused...
Why is it between 0 and 9?
 
  • #6
JC3187 said:
Still am confused...
Why is it between 0 and 9?
Because, for the natural log to be defined, x2 + y2 < 9,
but the smallest value of x2 + y2 is 0.
Therefore, you have 0 ≤ x2 + y2 < 9
 
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  • #7
Things are now clear, Thank you all!
 

FAQ: Finding the range of this function of 2 variables:

What is the definition of a function of 2 variables?

A function of 2 variables is a mathematical rule that assigns a unique output to every combination of two input values. It can be represented by an equation in the form of f(x,y) where x and y are the input variables and f(x,y) is the output.

How do you find the range of a function of 2 variables?

To find the range of a function of 2 variables, you need to plug in different values for the input variables and observe the corresponding output values. The set of all output values is the range of the function. Alternatively, you can also graph the function and identify the vertical spread of the graph, which represents the range.

What is the importance of finding the range of a function of 2 variables?

Finding the range of a function of 2 variables is important because it helps us understand the behavior of the function and its relationship between the input and output. It also allows us to identify the maximum and minimum values of the function and helps in solving optimization problems.

What are some common methods for finding the range of a function of 2 variables?

Some common methods for finding the range of a function of 2 variables include substitution, graphing, and setting up and solving inequalities. These methods rely on identifying the output values for different combinations of input values.

Are there any limitations to finding the range of a function of 2 variables?

Yes, there can be limitations to finding the range of a function of 2 variables. In some cases, the range may be infinite or undefined. Also, certain functions may have multiple ranges or no range at all. It is important to carefully consider the domain and the behavior of the function before attempting to find its range.

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