In Exercises 31–50, find ƒ-g and determine the domain for each function. f(x) = 2 + 1/x, g(x) = 1/x

Key Concepts

Function Subtraction

Function subtraction involves taking two functions, f(x) and g(x), and creating a new function, ƒ-g, defined as ƒ(x) - g(x). This operation requires combining the outputs of both functions for the same input value, which can lead to new expressions that may have different properties than the original functions.

Domain of a Function

The domain of a function is the set of all possible input values (x-values) for which the function is defined. For rational functions like f(x) = 2 + 1/x and g(x) = 1/x, the domain excludes values that make the denominator zero, as these would result in undefined outputs.

Combining Domains

When subtracting two functions, the domain of the resulting function ƒ-g is determined by the intersection of the domains of f(x) and g(x). This means that any x-value that is not in the domain of either function cannot be included in the domain of the new function, ensuring that ƒ-g is defined for those inputs.