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A composite function is essentially one function applied after another. To construct a composite function, you take two functions, say f and g, and apply them sequentially. This is denoted as f \( \circ \) g, where g is applied first, and its output is then used as the input for function f. In the given exercise, we have functions f(t) and g(x, y), where f(t) operates on a single variable, and g(x, y) on two variables, to form the composite function f(g(x, y)).

Understanding composite functions is crucial because they can model complex situations where one action depends on the outcome of another. For instance, changes in air temperature can affect air pressure, which then affects the boiling point of water. In our exercise, the composite function f \( \circ \) g(x, y) represents a more intricate relationship than either f or g alone.

The domain of a function includes all possible input values, while the range is the set of all possible output values. For our composite function f \( \circ \) g, understanding both functions' domains is vital. The function g(x, y) = x^2 + y^2 has a domain of all real number pairs for x and y, since any real x and y values provide a real output. On the other hand, the function f(t) = \frac{1}{4-t} has a domain of all real numbers except for t = 4.

The composite function's domain will be restricted by both of the original functions. The domain of f \( \circ \) g excludes any (x, y) pair where the output of g(x, y) would result in a t that is not part of the domain of f. Hence, f \( \circ \) g(x, y) is undefined for any (x, y) where x^2 + y^2 = 4, since that would feed a 4 into function f, which is not allowed. Hence, f \( \circ \) g has a discontinuity there.

Discontinuities in functions occur where functions are not continuous; these points are where there are jumps, holes, or breaks in the graph of the function. For our composite function f \( \circ \) g(x, y), discontinuities would arise at the points not within the domain of the composite function. Since f(t) is undefined at t=4, any (x, y) that causes g(x, y) to be 4 will create a discontinuity in the composite function. In our particular case, this is when the output of g, which is x^2 + y^2, equals 4.

This specific type of discontinuity is called a removable discontinuity because if we were to 'remove' those (x, y) values from the domain, we would have a perfectly continuous function everywhere else. If graphed in three dimensions, we would see a surface interrupted by a circular hole where x^2 + y^2 = 4. Identifying and understanding these discontinuities is important for both calculating accurate results from functions and also for understanding behavior around critical points.