Final answer:
The domain of the function f(x, y) = cos(ln(x² + y² - 25)) is all points (x, y) such that x² + y² - 25 > 0, representing a circle of radius 5 centered at the origin, excluding its boundary. The function is defined under these conditions due to properties of logarithmic and cosine operations.
Step-by-step explanation:
The subject of this question is the domain of a function involving a combination of the cosine, logarithm, and square root operations. The function is given as f(x, y) = cos(ln(x² + y² - 25)). A domain of a function consists of all the possible input values for which the function is defined.
To determine the domain for this function, we have to satisfy two conditions because of the cos(ln(...)) structure of the function. First, the argument of the logarithm (x² + y² - 25) must be greater than zero, because the logarithm isn't defined for negative numbers or zero. Second, the cosine function is defined for all real numbers, so we don't get an additional restriction from this.
Thus, the domain of the function is all points (x, y) such that x² + y² - 25 > 0. This represents a circle of radius 5 centered at the origin in the xy-plane, excluding the boundary of the circle (as the boundary would give us x² + y² - 25 = 0, which is not allowed). So, to sketch this domain, you would draw a circle with radius 5, centered at the origin, with the area inside the circle excluding the boundary.
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