<p>The range of the function <i>f</i>(<i>x</i>) = sec<sup>−1</sup>(<i>x</i>) + tan<sup>−1</sup>(<i>x</i>) is</p>
<p>(a) (0, π)</p>
<p>(b) [\(\frac{π}{12}\), \(\frac{π}{2}\)]</p>
<p>(c) (0, \(\frac{3π}{4}\))</p>
<p>(d) None of these</p>
Step-by-Step Solution
Key Concept: Determine the domain first, then analyze the range of each component function over the domain to find the overall range.
<p><strong>Step 1:</strong> The domain of <i>f</i>(<i>x</i>) is: <i>D<sub>f</sub></i> = (−∞, −1] ∪ [1, ∞)</p><p><strong>Step 2:</strong> For <i>x</i> ≥ 1: sec<sup>−1</sup>(<i>x</i>) ∈ [0, $\frac{π}{2}$) and tan<sup>−1</sup>(<i>x</i>) ∈ ($\frac{π}{4}$, $\frac{π}{2}$)</p><p><strong>Step 3:</strong> For <i>x</i> ≤ −1: sec<sup>−1</sup>(<i>x</i>) ∈ ($\frac{π}{2}$, π] and tan<sup>−1</sup>(<i>x</i>) ∈ (−$\frac{π}{2}$, −$\frac{π}{4}$)</p><p><strong>Step 4:</strong> Analyzing the behavior at boundary points and over the domain, the range does not match any of options (a), (b), or (c).</p><p>∴ Answer is (d) None of these</p>
Correct Answer: D