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

검증된 단계별 안내

Step 1: Understand the problem. You are tasked with finding the sum of two functions, f(x) and g(x), which is denoted as (f + g)(x). This means you need to add the two given functions together: f(x) = 2 + 1/x and g(x) = 1/x.

Step 2: Write the expression for (f + g)(x). Add the two functions together: (f + g)(x) = f(x) + g(x) = (2 + 1/x) + (1/x).

Step 3: Combine like terms. Simplify the expression by combining the terms involving 1/x: (f + g)(x) = 2 + 1/x + 1/x = 2 + 2/x.

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 this problem, note that both f(x) and g(x) contain the term 1/x, which is undefined when x = 0. Therefore, the domain of (f + g)(x) excludes x = 0.

Step 5: Express the domain in interval notation. Since x = 0 is excluded, the domain is all real numbers except 0. In interval notation, this is written as (-∞, 0) ∪ (0, ∞).

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주요 개념

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Function Addition

Function addition involves combining two functions by adding their outputs for each input. For functions f(x) and g(x), the sum is defined as (f + g)(x) = f(x) + g(x). This operation requires evaluating both functions at the same input value and summing the results.

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.

Rational Functions

Rational functions are ratios of polynomials, expressed in the form f(x) = P(x)/Q(x), where P and Q are polynomials. The behavior of rational functions, including their domains and asymptotes, is influenced by the zeros of the denominator, which dictate where the function is undefined.