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Chapter 11: Problem 22

Describe the domain and range of the function. $$ f(x, y)=\ln (x y-6) $$

Short Answer

Expert verified
The domain of the function \(f(x, y)=\ln(xy-6)\) consists of all pairs of real numbers \((x, y)\) such that the product of x and y is greater than 6. Hence, \(D_f = {(x,y) \in R^2 | xy > 6}\). The range of the function is all real numbers, i.e., \(R_f = R\).

Step by step solution

01

Identify the Inequality

Note that for \(f(x, y)=\ln(xy-6)\) to be defined, \(xy-6\) must be greater than 0. This leads to a simple inequality to be solved: \(xy-6 > 0\)
02

Solve the Inequality

Solve the inequality for xy, yielding \(xy > 6\). This implies that the product of x and y must be greater than 6.
03

Establish the Domain

The domain (values for x and y) are thus all pairs of real numbers \(x, y\) such that the product of x and y is greater than 6.
04

Determine the Range

Regarding the range, note that the natural logarithm function, \(\ln(x)\), covers all real numbers for \(x > 0\), so once \(xy-6\) is greater than zero, the range of \(f(x, y)\) will be all real numbers.

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