Find ƒ-g and determine the domain for each function. f(x) = 2 + 1/x, g(x) = 1/x

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Step 1: Understand the problem. You are tasked with finding the difference of two functions, ƒ(x) and g(x), which is represented as (ƒ - g)(x). This means you need to subtract g(x) from ƒ(x). The functions are given as ƒ(x) = 2 + 1/x and g(x) = 1/x.

Step 2: Write the expression for (ƒ - g)(x). Subtract g(x) from ƒ(x): (ƒ - g)(x) = ƒ(x) - g(x). Substituting the given functions, this becomes (ƒ - g)(x) = (2 + 1/x) - (1/x).

Step 3: Simplify the expression. Combine like terms. Notice that the terms involving 1/x cancel out: (ƒ - g)(x) = 2 + 1/x - 1/x = 2.

Step 4: Determine the domain of the resulting function. The domain of a function is the set of all x-values for which the function is defined. For the original functions ƒ(x) and g(x), the term 1/x implies that x cannot be 0 (division by zero is undefined). Therefore, the domain of (ƒ - g)(x) is all real numbers except x = 0.

Step 5: Write the final result. The simplified function is (ƒ - g)(x) = 2, and the domain is all real numbers except x = 0, which can be expressed as (-∞, 0) ∪ (0, ∞) in interval notation.

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주요 개념

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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, such as domain and range.

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 any 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.