Trigonometry & Inverse Trigonometry
Function Properties with Inverse Trigonometry
Grade 12
Question:
<p>Let \(g: \mathbb{R} \to \left[0, \frac{7\pi}{2}\right)\) is defined by \(g(x) = \cos^{-1}\frac{x}{1+x^2}\). Then the possible values of \(k\) for which \(g\) is a surjective function, is</p>
<p>(a) \(k \in [-1, 1]\)</p>
<p>(b) \(k \in \left[-\frac{1}{2}, \frac{1}{2}\right]\)</p>
<p>(c) \(k \in [-1, 1]\)</p>
<p>(d) \(k \in [-1, 1]\)</p>
Step-by-Step Solution
Key Concept: For surjectivity, the range of $g(x) = \cos^{-1}\frac{x}{1+x^2}$ must equal $[0, \frac{7\pi}{2})$; find the range by analyzing $\frac{x}{1+x^2}$.
<p>No solution provided in source text</p>
Correct Answer: A