Textbook Question
Verify the integration formulas in Exercises 111–114.
111. ∫ (arctan x) / x² dx = ln x - 1/2 ln(1 + x²) - arctan x / x + C
Ch. 7 - Transcendental 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 Functions490
- Ch. 8 - Techniques of Integration399
- Ch. 9 - First-Order Differential Equations66
- Ch. 10 - Infinite Sequences and Series132
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 7, Problem 7.1.63
Suppose that the range of g lies in the domain of f so that the composition fog is defined. If f and g are one-to-one, can anything be said about fog? Give reasons for your answer.
Verified step by step guidance1
Recall the definition of a one-to-one (injective) function: a function \( h \) is one-to-one if \( h(a) = h(b) \) implies \( a = b \).
Given that \( f \) and \( g \) are both one-to-one, consider the composition \( f \circ g \), defined by \( (f \circ g)(x) = f(g(x)) \).
To check if \( f \circ g \) is one-to-one, assume \( (f \circ g)(x_1) = (f \circ g)(x_2) \). This means \( f(g(x_1)) = f(g(x_2)) \).
Since \( f \) is one-to-one, \( f(g(x_1)) = f(g(x_2)) \) implies \( g(x_1) = g(x_2) \).
Because \( g \) is also one-to-one, \( g(x_1) = g(x_2) \) implies \( x_1 = x_2 \). Therefore, \( f \circ g \) is one-to-one.
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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 applying one function to the result of another, denoted as (f ∘ g)(x) = f(g(x)). For composition to be defined, the range of g must lie within the domain of f. Understanding this ensures the combined function is valid and can be analyzed.
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One-to-One (Injective) Functions
A function is one-to-one if it maps distinct inputs to distinct outputs, meaning no two different inputs share the same output. This property is crucial for invertibility and affects how compositions behave, especially regarding uniqueness of outputs.
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Injectivity of Compositions
The composition of two one-to-one functions is also one-to-one. Since both f and g are injective, their composition f ∘ g preserves distinctness of inputs, ensuring that (f ∘ g)(x1) ≠ (f ∘ g)(x2) whenever x1 ≠ x2, thus making f ∘ g injective.
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Related Practice
Textbook Question
In Exercises 7–26, find the derivative of y with respect to x, t, or θ, as appropriate.
y = e^(cost+lnt)
Textbook Question
132. What is special about the functions
f(x) = arcsin((1/√(x²+1)) and g(x)=arctan(1/x)?
Explain.
Textbook Question
Use l’Hôpital’s rule to find the limits in Exercises 7–52.
35. lim (x → 0⁺) ln(x² + 2x) / ln x
1
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Textbook Question
135. Find the area of the “triangular” region in the first quadrant that is bounded above by the curve y = e^(2x), below by the curve y = e^x, and on the right by the line x = ln(3).
Textbook Question
Each of Exercises 19–24 gives a formula for a function y=f(x) and shows the graphs of f and f^(-1). Find a formula for f^(-1) in each case.
f(x)=(x+1)², x≥-1