Ch. 2 - Functions and Graphs

Blitzer8th EditionCollege AlgebraISBN: 9780136970514Non è quello che usi tu?Cambia libro di testo

Blitzer 8th Edition

Ch. 2 - Functions and Graphs

Problema 41b

Capitolo 3, Problema 41b

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

Guida verificata passo dopo passo

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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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.

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Adding & Subtracting 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 any values that make the denominator zero, as these would result in undefined outputs.

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Domain Restrictions of Composed Functions

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.

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Combinations

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