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AP Precalculus -3.9 Inverse Trigonometric Functions- MCQ Exam Style Questions - Effective Fall 2023

AP Precalculus -3.9 Inverse Trigonometric Functions- MCQ Exam Style Questions – Effective Fall 2023

AP Precalculus -3.9 Inverse Trigonometric Functions- MCQ Exam Style Questions – AP Precalculus- per latest AP Precalculus Syllabus.

AP Precalculus – MCQ Exam Style Questions- All Topics

Question 

The function \( f \) is given by \( f(x) = \frac{1}{2} \sin x \) for \( -\frac{\pi}{2} \leq x \leq \frac{\pi}{2} \). What are the domain and range of the inverse function of \( f \)?
(A) Domain: \([-\frac{1}{2}, \frac{1}{2}]\), Range: \([-\frac{\pi}{2}, \frac{\pi}{2}]\)
(B) Domain: \([-1, 1]\), Range: \([-\frac{\pi}{2}, \frac{\pi}{2}]\)
(C) Domain: \([-\frac{\pi}{2}, \frac{\pi}{2}]\), Range: \([-\frac{1}{2}, \frac{1}{2}]\)
(D) Domain: \([-\frac{\pi}{2}, \frac{\pi}{2}]\), Range: \([-1, 1]\)
▶️ Answer/Explanation
Detailed solution

For the original function \( f(x) = \frac{1}{2}\sin x \) on \( -\frac{\pi}{2} \leq x \leq \frac{\pi}{2} \):
– Domain of \( f \): \( [-\frac{\pi}{2}, \frac{\pi}{2}] \)
– Range of \( f \): Since \(\sin x\) ranges from \(-1\) to \(1\) over this interval, \( \frac{1}{2}\sin x \) ranges from \(-\frac{1}{2}\) to \(\frac{1}{2}\).
For the inverse function \( f^{-1} \), the domain and range swap:
– Domain of \( f^{-1} \) = Range of \( f \) = \([-\frac{1}{2}, \frac{1}{2}]\)
– Range of \( f^{-1} \) = Domain of \( f \) = \([-\frac{\pi}{2}, \frac{\pi}{2}]\)
✅ Answer: (A)

Question 

 
 
 
 
 
 
 
 
 
 
 
 
The graph of a sinusoidal function \( f \) is given. What is the length of the largest interval of input values of \( f \) on which an inverse function of \( f \) can be constructed?
(A) 1
(B) 2
(C) 4
(D) There is no interval larger than a single point on which \( f \) is invertible.
▶️ Answer/Explanation
Detailed solution

For a sinusoidal function to be invertible, it must be one-to-one on an interval. The longest such interval on a sine or cosine wave is half a period where the function is strictly increasing or decreasing.
From the graph (described in the PDF), the period is 4, so half-period is 2. This gives the longest interval for an inverse.
✅ Answer: (B)

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