Ch. 2 - Functions and Graphs

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

Blitzer 8th Edition

Ch. 2 - Functions and Graphs

Problem 41b

Chapter 3, Problem 41b

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 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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Key Concepts

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

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.

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Combinations

Related Practice

Textbook Question

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

Textbook Question

Find $f+g$ , $f-g$ , $fg$ , and $\frac{f}{g}$ . Determine the domain for each function.

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Textbook Question

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

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Textbook Question

In Exercises 39-52, a. Find an equation for ƒ¯¹(x). b. Graph ƒ and ƒ¯¹(x) in the same rectangular coordinate system. c. Use interval notation to give the domain and the range of f and ƒ¯¹. ƒ(x) = x² − 4, x ≥ 0

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Textbook Question

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

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