Textbook Question
Find the largest interval on which the given function is increasing.
d. R(x) = √ 2x - 1
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Ch. 1 - Functions
Hass15th EditionThomas' CalculusISBN: 9780137616077
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Table of contents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions489
- Ch. 8 - Techniques of Integration398
- Ch. 9 - First-Order Differential Equations60
- Ch. 10 - Infinite Sequences and Series119
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 1, Problem 39a
Composition of Functions
In Exercises 39 and 40, find
a. (ƒ ○ g) (-1).
ƒ(x) = 1/x , g(x) = 1/√ x + 2
Verified step by step guidance1
Understand the composition of functions: The composition (ƒ ○ g)(x) means applying g first and then applying ƒ to the result of g(x).
Substitute x = -1 into the function g(x): g(-1) = 1/√(-1 + 2). Calculate the expression inside the square root first.
Simplify the expression inside g(-1): √(-1 + 2) = √1 = 1. Therefore, g(-1) = 1/1 = 1.
Now apply the function ƒ to the result of g(-1): ƒ(g(-1)) = ƒ(1). Substitute 1 into ƒ(x) = 1/x.
Simplify ƒ(1): ƒ(1) = 1/1 = 1. Therefore, (ƒ ○ g)(-1) = 1.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Function Composition
Function composition involves combining two functions to create a new function. The notation (ƒ ○ g)(x) means to apply function g first and then apply function f to the result of g. This is essential for solving the problem, as it requires evaluating g at a specific input and then using that output as the input for f.
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3:48
Evaluate Composite Functions - Special Cases
Evaluating Functions
Evaluating a function means substituting a specific value into the function's formula to find the output. In this case, we need to evaluate g at -1, which requires substituting -1 into the function g(x) = 1/√(x + 2). Understanding how to correctly substitute values is crucial for finding the correct outputs in function composition.
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4:26Evaluating Composed Functions
Domain of Functions
The domain of a function is the set of all possible input values for which the function is defined. For the given functions, we must consider the domains of both f(x) = 1/x and g(x) = 1/√(x + 2) to ensure that the inputs do not lead to undefined expressions, such as division by zero or taking the square root of a negative number.
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5:10Finding the Domain and Range of a Graph
Related Practice
Textbook Question
Graph ƒ₁ and ƒ₂ together. Then describe how applying the absolute value function in ƒ₂ affects the graph of ƒ₁.
ƒ₁(x) ƒ₂(x)
x² |x|²
Textbook Question
State whether each function is increasing, decreasing, or neither.
d. Kinetic energy as a function of a particle’s velocity
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Textbook Question
In Exercises 41 and 42, (a) write formulas for ƒ ○ g and g ○ ƒ and find the (b) domain and (c) range of each.
ƒ(x) = 2 - x², g(x) = √ x + 2
Textbook Question
Composition of Functions
In Exercises 39 and 40, find
d. (g ○ g) (x).
ƒ(x) = 1/x , g(x) = 1/√ x + 2
Textbook Question
Find the largest interval on which the given function is increasing.
a. ƒ(x) = |x - 2| + 1
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