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Ch. 1 - Functions
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 1 - FunctionsProblem 1.2.2
Chapter 1, Problem 1.2.2

Algebraic Combinations


In Exercises 1 and 2, find the domains of f, g, f + g, and f ⋅ g.


f(x) = √(x + 1), g(x) = √(x − 1)

Verified step by step guidance
1
Step 1: Determine the domain of f(x) = √(x + 1). The expression inside the square root, x + 1, must be greater than or equal to zero for the square root to be defined. Solve the inequality x + 1 ≥ 0 to find the domain of f(x).
Step 2: Determine the domain of g(x) = √(x − 1). Similarly, the expression inside the square root, x − 1, must be greater than or equal to zero. Solve the inequality x − 1 ≥ 0 to find the domain of g(x).
Step 3: Find the domain of the sum f + g. The domain of f + g is the intersection of the domains of f and g, as both functions must be defined for the sum to be defined.
Step 4: Find the domain of the product f ⋅ g. The domain of f ⋅ g is also the intersection of the domains of f and g, since both functions must be defined for the product to be defined.
Step 5: Combine the results from Steps 1 to 4 to express the domains of f, g, f + g, and f ⋅ g in interval notation.

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Key Concepts

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

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 functions involving square roots, such as f(x) = √(x + 1) and g(x) = √(x − 1), the expressions under the square roots must be non-negative. This means we need to find the values of x that satisfy these conditions to determine the domain.
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Algebraic Combinations of Functions

Algebraic combinations of functions involve operations such as addition, subtraction, multiplication, and division applied to two or more functions. In this case, we are looking at f + g and f ⋅ g. The domain of these combinations is determined by the intersection of the individual domains of the functions involved, ensuring that the resulting expressions are defined.
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Square Root Function Properties

The square root function has specific properties that affect its domain and range. The output of a square root function is always non-negative, and the input must be greater than or equal to zero. Understanding these properties is crucial for determining the domains of f(x) and g(x), as they dictate the restrictions on x that must be considered when combining these functions.
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Related Practice
Textbook Question

Increasing and Decreasing Functions


Graph the functions in Exercises 37–46. What symmetries, if any, do the graphs have? Specify the intervals over which the function is increasing and the intervals where it is decreasing.


y = −x³

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Textbook Question

Use graphing software to graph the functions specified in Exercises 31–36.

Select a viewing window that reveals the key features of the function.


Graph the function f (x) = sin 2x + cos 3x.

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Textbook Question

Using the Addition Formulas


Use the addition formulas to derive the identities in Exercises 31–36.


sin (A − B) = sin A cos B − cos A sin B

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Textbook Question

Increasing and Decreasing Functions


Graph the functions in Exercises 37–46. What symmetries, if any, do the graphs have? Specify the intervals over which the function is increasing and the intervals where it is decreasing.


y = (−x)²/³

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Textbook Question

[Technology Exercise]


a. Graph the functions f(x) = x/2 and g(x) = 1 + (4/x) together to identify the values of x for which


x/2 > 1 + 4/x


b. Confirm your findings in part (a) algebraically.

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Textbook Question

In Exercises 5–8, determine whether the graph of the function is symmetric about the 𝔂-axis, the origin, or neither.


𝔂 = e⁻ˣ²

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