Question

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

Accepted Answer

Identify the base function given, which is the inverse cosine function $$y = \cos$$^{-1}$$ x. $$This function has a domain of $$[-1, 1]$$ and a range of $$[0, \pi]. $$Recognize that the function provided is $$f(x) = \cos$$^{-1}$$ \left( \frac{x}{2} \right)$$, which means the input to the inverse cosine function is transformed by dividing $$x by 2. $$Determine the domain of $$f(x) by $$setting the inside of the inverse cosine function within its valid domain: $$-1 \leq \frac{x}{2} \leq 1. $$Solve this inequality for $$x to $$find the domain of $$f(x). $$Understand that the range of $$f(x)$$ remains the same as the range of $$\cos$$^{-1}$$ x$$, because the transformation is inside the function and does not affect the output values. So, the range is $$[0, \pi]. To $$graph $$f(x)$$, start with the graph of $$y = \cos$$^{-1}$$ x$$ and apply a horizontal stretch by a factor of 2 (since $$x is $$divided by 2 inside the function). This means the graph will be stretched horizontally, making it wider.

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Key Concepts

Inverse Trigonometric Functions

Function Transformations

Domain and Range of Transformed Functions