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

Verified step by step guidance

1. Identify the base function: The base function here is $y = \tan^{-1}(x)$, which is the inverse tangent function.
2. Apply the transformation: The given function is $h(x) = -2 \tan^{-1}(x)$. This involves two transformations: a vertical reflection and a vertical stretch.
3. Vertical reflection: The negative sign in front of $2 \tan^{-1}(x)$ indicates a reflection across the x-axis.
4. Vertical stretch: The coefficient 2 indicates a vertical stretch by a factor of 2. This means that the graph will be stretched away from the x-axis.
5. Determine the domain and range: The domain of $\tan^{-1}(x)$ is all real numbers, $(-\infty, \infty)$. The range of $\tan^{-1}(x)$ is $\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$. After the transformations, the range of $h(x)$ becomes $\left(-\pi, \pi\right)$.

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions, such as sin⁻¹, cos⁻¹, and tan⁻¹, are the functions that reverse the action of the standard trigonometric functions. For example, if y = sin(x), then x = sin⁻¹(y). These functions are defined for specific ranges of their respective trigonometric functions, which is crucial for understanding their graphs and transformations.

Transformations of Functions

Transformations of functions involve altering the graph of a function through various operations, including vertical shifts (up or down), horizontal shifts (left or right), reflections (over axes), and stretching or shrinking (changing the scale). Understanding these transformations is essential for accurately graphing modified functions, such as h(x) = −2 tan⁻¹ x, based on the original inverse trigonometric function.

Domain and Range

The domain of a function refers to the set of all possible input values (x-values) for which the function is defined, while the range refers to the set of all possible output values (y-values). For inverse trigonometric functions, the domain and range are limited to specific intervals, which must be considered when applying transformations. For example, the domain of tan⁻¹ x is all real numbers, while its range is (-π/2, π/2).