Limits, Continuity & Differentiability
Continuity of composite and rational functions
Grade 12
Question:
<p>Let <span>\([x]\)</span> be the greatest integer function. Indicate the correct alternative, if <span>\(f(x) = x^2 + 1\)</span>, then on the interval <span>\([0, \pi]\)</span>, <span>\(\tan(f(x))\)</span> and <span>\(\frac{1}{f(x)}\)</span> are:</p>
<p>(a) both continuous</p>
<p>(b) both discontinuous</p>
<p>(c) <span>\(\tan(f(x))\)</span> and <span>\(f^{-1}(x)\)</span> are both continuous</p>
<p>(d) <span>\(\tan(f(x))\)</span> is continuous but <span>\(\frac{1}{f(x)}\)</span> is not continuous</p>
Step-by-Step Solution
Key Concept: Continuity of composite functions and rational functions depends on the continuity of component functions and nonzero denominators.
<p><strong>Analysis:</strong> Since <span>$f(x) = x^2 + 1$</span> is a polynomial, it is continuous everywhere. The function <span>$\tan(f(x))$</span> is continuous wherever <span>$f(x) \neq \frac{\pi}{2} + n\pi$</span>. On <span>$[0, \pi]$</span>, we have <span>$1 \leq f(x) \leq \pi^2 + 1$</span>, so <span>$\tan(f(x))$</span> is continuous. Also <span>$\frac{1}{f(x)}$</span> is continuous since <span>$f(x) > 0$</span> for all <span>$x \in [0, \pi]$</span>.</p><p>∴ Answer is (a).</p>
Correct Answer: A