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Ch. 2 - Graphs of the Trigonometric Functions; Inverse Trigonometric Functions
Blitzer - Trigonometry 3rd Edition
Blitzer3rd EditionTrigonometryISBN: 9780137316601Not the one you use?Change textbook
All textbooksBlitzer 3rd EditionCh. 2 - Graphs of the Trigonometric Functions; Inverse Trigonometric FunctionsProblem 101
Chapter 2, Problem 101

The graphs of y = sin⁻¹ x, y = cos⁻¹ x, and y = tan⁻¹ x are shown in Table 2.8. In Exercises 97–106, use transformations (vertical shifts, horizontal shifts, reflections, stretching, or shrinking) of these graphs to graph each function. Then use interval notation to give the function's domain and range. h(x) = −2 tan⁻¹ x

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Identify the base function given, which is the inverse tangent function \(y = \tan^{-1} x\). Recall that its standard domain is \((-\infty, \infty)\) and its range is \(\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)\).
Analyze the transformation applied to the base function. The function is \(h(x) = -2 \tan^{-1} x\), which involves two transformations: a vertical stretch by a factor of 2 and a reflection about the x-axis (due to the negative sign).
Apply the vertical stretch and reflection to the range of \(\tan^{-1} x\). Since the original range is \(\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)\), multiplying by -2 will stretch the range by 2 and flip it, so the new range will be \(\left(-2 \times \frac{\pi}{2}, -2 \times -\frac{\pi}{2}\right)\).
Note that the domain of \(\tan^{-1} x\) is all real numbers, and since the transformation only affects the output (y-values), the domain of \(h(x)\) remains \((-\infty, \infty)\).
Summarize the domain and range in interval notation after applying the transformations, and sketch the graph by reflecting and stretching the original \(\tan^{-1} x\) graph accordingly.

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Key Concepts

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

Inverse Trigonometric Functions

Inverse trigonometric functions like sin⁻¹x, cos⁻¹x, and tan⁻¹x return the angle whose trigonometric ratio equals x. They have specific domains and ranges, for example, tan⁻¹x has domain all real numbers and range (−π/2, π/2). Understanding these functions is essential for graphing and interpreting their transformations.
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Introduction to Inverse Trig Functions

Graph Transformations

Graph transformations include vertical and horizontal shifts, reflections, stretches, and shrinks. For h(x) = −2 tan⁻¹ x, the negative sign reflects the graph across the x-axis, and the factor 2 vertically stretches it by a factor of 2. Recognizing how these changes affect the graph helps in sketching and analyzing the function.
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Introduction to Transformations

Domain and Range of Transformed Functions

The domain of inverse trig functions often remains unchanged under vertical transformations, but the range changes according to vertical stretches or reflections. For h(x) = −2 tan⁻¹ x, the domain is all real numbers, while the range is the original range (−π/2, π/2) scaled and reflected, resulting in (−π, π). Using interval notation accurately describes these sets.
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Domain and Range of Function Transformations
Related Practice
Textbook Question

In Exercises 83–94, use a right triangle to write each expression as an algebraic expression. Assume that x is positive and that the given inverse trigonometric function is defined for the expression in x. csc (cot⁻¹ x)

Textbook Question

The graphs of y = sin⁻¹ x, y = cos⁻¹ x, and y = tan⁻¹ x are shown in Table 2.8. In Exercises 97–106, use transformations (vertical shifts, horizontal shifts, reflections, stretching, or shrinking) of these graphs to graph each function. Then use interval notation to give the function's domain and range. f(x) = sin⁻¹ x + π/2

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Textbook Question

The graphs of y = sin⁻¹ x, y = cos⁻¹ x, and y = tan⁻¹ x are shown in Table 2.8. In Exercises 97–106, use transformations (vertical shifts, horizontal shifts, reflections, stretching, or shrinking) of these graphs to graph each function. Then use interval notation to give the function's domain and range. f(x) = cos⁻¹ x/2

Textbook Question

The graphs of y = sin⁻¹ x, y = cos⁻¹ x, and y = tan⁻¹ x are shown in Table 2.8. In Exercises 97–106, use transformations (vertical shifts, horizontal shifts, reflections, stretching, or shrinking) of these graphs to graph each function. Then use interval notation to give the function's domain and range. f(x) = cos⁻¹ (x + 1)

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