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

Verified step by step guidance

1. Step 1: Understand the base function. The base function here is \( y = \cos^{-1}(x) \), which is the inverse cosine function. Its domain is \([-1, 1]\) and its range is \([0, \pi]\).
2. Step 2: Identify the transformation. The given function is \( f(x) = \cos^{-1}(\frac{x}{2}) \). This represents a horizontal stretch of the base function by a factor of 2.
3. Step 3: Determine the new domain. Since the transformation involves \( \frac{x}{2} \), solve \( -1 \leq \frac{x}{2} \leq 1 \) to find the new domain. Multiply through by 2 to get \(-2 \leq x \leq 2\).
4. Step 4: Determine the range. The range of the inverse cosine function remains unchanged by horizontal stretching, so the range is still \([0, \pi]\).
5. Step 5: Graph the function. Use the transformations identified to sketch the graph of \( f(x) = \cos^{-1}(\frac{x}{2}) \) based on the graph of \( y = \cos^{-1}(x) \), applying the horizontal stretch.

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions, such as sin⁻¹, cos⁻¹, and tan⁻¹, are used to find angles when given a ratio of sides in a right triangle. Each function has a specific range and domain, which are crucial for understanding their graphs. For example, the range of cos⁻¹ x is [0, π], meaning it outputs angles between 0 and π radians.

Transformations of Functions

Transformations involve altering the graph of a function through shifts, stretches, or reflections. For instance, a vertical shift can move the graph up or down, while a horizontal shift moves it left or right. Understanding these transformations is essential for accurately graphing modified functions like f(x) = cos⁻¹(x/2).

Domain and Range

The domain of a function refers to all possible input values (x-values), while the range refers to all possible output values (y-values). For inverse trigonometric functions, the domain is typically restricted to ensure the function is defined. For f(x) = cos⁻¹(x/2), determining the domain involves finding the values of x that keep the argument within the valid range of the original cosine function.