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Ch. 7 - Transcendental Functions
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 7 - Transcendental FunctionsProblem 7.6.139a
Chapter 7, Problem 7.6.139a

[Technology Exercise] In Exercises 139–141, find the domain and range of each composite function. Then graph the compositions on separate screens. Do the graphs make sense in each case? Give reasons for your answers. Comment on any differences you see.
139. a. y=arctan(tan x)

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Identify the inner and outer functions in the composition. Here, the inner function is \(\tan x\) and the outer function is \(\arctan y\), where \(y = \tan x\).
Recall the domain and range of each function separately: \(\tan x\) has domain \(x \neq \frac{\pi}{2} + k\pi\), \(k \in \mathbb{Z}\), and range \((-\infty, \infty)\); \(\arctan x\) has domain \((-\infty, \infty)\) and range \(\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)\).
Since \(\arctan\) is applied to \(\tan x\), the domain of the composite function \(y = \arctan(\tan x)\) is the domain of \(\tan x\), which excludes points where \(\tan x\) is undefined.
The range of the composite function is limited by the range of \(\arctan\), so \(y\) will lie in \(\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)\).
Consider the behavior of \(\arctan(\tan x)\) over intervals where \(\tan x\) is continuous and increasing. The function essentially 'wraps' \(x\) back into the principal range of \(\arctan\), causing a periodic 'sawtooth' pattern. This explains why the graph repeats and why the range is restricted.

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Key Concepts

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

Composite Functions

A composite function is formed when one function is applied to the result of another, written as (f ∘ g)(x) = f(g(x)). Understanding how to combine functions and evaluate their outputs is essential for analyzing the domain and range of compositions.
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Evaluate Composite Functions - Special Cases

Domain and Range of Trigonometric and Inverse Trigonometric Functions

The domain is the set of input values for which a function is defined, and the range is the set of possible output values. For example, tan(x) is defined for all real x except odd multiples of π/2, while arctan(x) has a range of (-π/2, π/2). Knowing these restrictions helps determine the domain and range of their composition.
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Derivatives of Other Inverse Trigonometric Functions

Graphing and Interpretation of Function Behavior

Graphing composite functions helps visualize their behavior, including periodicity, discontinuities, and range limitations. Comparing the graph of y = arctan(tan x) with the original functions reveals how the composition behaves and whether the outputs align with expected ranges.
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Graph of Sine and Cosine Function
Related Practice
Textbook Question

Evaluate the integrals in Exercises 67–74 in terms of

a. inverse hyperbolic functions.

67. ∫(from 0 to 2√3)dx/√(4+x²)

Textbook Question

In Exercises 41–44:

a. Find f⁻¹(x).


42. f(x) = (x + 2) / (1 − x), a = 1/2

Textbook Question

153. The linearization of 2ˣ

a. Find the linearization of f(x) = 2ˣ at x = 0. Then round its coefficients to two decimal places.

Textbook Question

75. a. Find the open intervals on which the function is increasing and decreasing.

g(x) = x(ln x)²

Textbook Question

145. The linearization of eˣ at x = 0

a. Derive the linear approximation eˣ ≈ 1 + x at x = 0.

Textbook Question

78. Which one is correct, and which one is wrong? Give reasons for your answers.

a. lim (x → 0) (x² - 2x) / (x² - sin x) = lim (x → 0) (2x - 2) / (2x - cos x) = lim (x → 0) 2 / (2 + sin x) = 2 / (2 + 0) = 1

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