Ch. 2 - Functions and Graphs

Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook

Blitzer 8th Edition

Ch. 2 - Functions and Graphs

Problem 41a

Chapter 3, Problem 41a

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

Verified step by step guidance

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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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.

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