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)

Verified step by step guidance

1. Identify the base function: The base function here is \( y = \cos^{-1}(x) \).
2. Determine the transformation: The function \( f(x) = \cos^{-1}(x + 1) \) involves a horizontal shift. The \( +1 \) inside the function indicates a shift to the left by 1 unit.
3. Graph the transformation: Start with the graph of \( y = \cos^{-1}(x) \), which is defined for \( x \in [-1, 1] \) and has a range of \( [0, \pi] \). Shift this graph 1 unit to the left to obtain the graph of \( f(x) = \cos^{-1}(x + 1) \).
4. Determine the new domain: The original domain of \( y = \cos^{-1}(x) \) is \( [-1, 1] \). After shifting left by 1 unit, the domain becomes \( [-2, 0] \).
5. Determine the range: The range of \( y = \cos^{-1}(x) \) is \( [0, \pi] \). Since the transformation is a horizontal shift, the range remains unchanged, \( [0, \pi] \).

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions, such as sin⁻¹, cos⁻¹, and tan⁻¹, are the inverses of the standard trigonometric functions. They are used to determine the angle whose sine, cosine, or tangent is a given value. Each function has a specific range and domain, which are crucial for understanding their graphs and transformations.

Transformations of Functions

Transformations involve altering the graph of a function through shifts, stretches, shrinks, or reflections. For example, adding a constant to the input (x) results in a horizontal shift, while adding a constant to the output (y) results in a vertical shift. Understanding these transformations is essential for accurately graphing modified functions like f(x) = cos⁻¹(x + 1).

Domain and Range

The domain of a function refers to the set of all possible input values (x-values), 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. For instance, the domain of cos⁻¹(x) is [-1, 1], and its range is [0, π], which must be considered when applying transformations.