Relations & Functions
Range of Functions
Grade 12
Question:
<p>Let \(f(x)\) is continuous function with range \([-1, 1]\) and \(f(x)\) is defined \(\forall x \in \mathbb{R}\). If \(g(x) = \frac{e^{f(x)} - e^{|f(x)|}}{e^{f(x)} + e^{|f(x)|}}\), then range of \(g(x)\) is:</p>
<p>(a) \([0, 1]\)</p>
<p>(b) \(\left[0, \frac{e^2+1}{e^2-1}\right]\)</p>
<p>(c) \(\left[0, \frac{e^2-1}{e^2+1}\right]\)</p>
<p>(d) \(\left[\frac{-e^2+1}{e^2+1}, 0\right]\)</p>
Step-by-Step Solution
Key Concept: Analyze \(g(x)\) by considering the cases \(f(x) \geq 0\) and \(f(x) < 0\). When \(f(x) \geq 0\), \(|f(x)| = f(x)\) so \(g(x) = 0\). When \(f(x) < 0\), simplify using \(|f(x)| = -f(x)\) to find the range.
<p>Solution not provided in source text.</p>
Correct Answer: c