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

Verified step by step guidance

1. Identify the base function: The base function here is \(y = \sin^{-1} x\), which is the inverse sine function.
2. Understand the transformation: The function \(f(x) = \sin^{-1} x + \frac{\pi}{2}\) involves a vertical shift. Specifically, it shifts the graph of \(y = \sin^{-1} x\) upward by \(\frac{\pi}{2}\) units.
3. Graph the transformation: Start by sketching the graph of \(y = \sin^{-1} x\). Then, shift every point on this graph upward by \(\frac{\pi}{2}\) units to obtain the graph of \(f(x) = \sin^{-1} x + \frac{\pi}{2}\). This means if a point on \(y = \sin^{-1} x\) is \((a, b)\), it will be \((a, b + \frac{\pi}{2})\) on \(f(x)\).
4. Determine the domain: The domain of \(y = \sin^{-1} x\) is \([-1, 1]\). Since the transformation is a vertical shift, the domain of \(f(x)\) remains unchanged, which is \([-1, 1]\).
5. Determine the range: The range of \(y = \sin^{-1} x\) is \([-\frac{\pi}{2}, \frac{\pi}{2}]\). After shifting the graph upward by \(\frac{\pi}{2}\), the range of \(f(x)\) becomes \([0, \pi]\).

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions, such as sin⁻¹ x, cos⁻¹ x, and tan⁻¹ x, are used to find angles when given a ratio of sides in a right triangle. These functions have specific domains and ranges: for example, sin⁻¹ x has a domain of [-1, 1] and a range of [-π/2, π/2]. Understanding these properties is crucial for graphing and transforming these functions.

Transformations of Functions

Transformations involve altering the graph of a function through vertical shifts, horizontal shifts, reflections, stretching, or shrinking. For instance, adding a constant to a function results in a vertical shift, while multiplying by a factor greater than one stretches the graph. Mastery of these transformations allows for the manipulation of the graphs of inverse trigonometric functions to create new functions.

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 the function f(x) = sin⁻¹ x + π/2, the domain remains [-1, 1], but the range shifts due to the vertical transformation, resulting in a new range of [π/2 - π/2, π/2 + π/2] or [0, π]. Understanding how transformations affect domain and range is essential for accurately describing the function.