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.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

10m

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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.

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.

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.