Limits, Continuity & Differentiability
Monotonicity
Grade 12
Question:
<p>If <i>f</i> : ℝ → ℝ is the function defined by <i>f</i>(<i>x</i>) = (e<sup><i>x</i></sup> − e<sup>−<i>x</i></sup>)/(e<sup><i>x</i></sup> + e<sup>−<i>x</i></sup>), then</p>
<p>(a) <i>f</i>(<i>x</i>) is an increasing function</p>
<p>(b) <i>f</i>(<i>x</i>) is a decreasing function</p>
<p>(c) <i>f</i>(<i>x</i>) is onto</p>
<p>(d) None of the above</p>
Step-by-Step Solution
Key Concept: To determine if f(x) is increasing, decreasing, or onto, we need to find f'(x) and analyze its sign, and also determine the range of f(x). The function f(x) = (eˣ - e⁻ˣ)/(eˣ + e⁻ˣ) is actually tanh(x), the hyperbolic tangent function.
<p><strong>Step 1: Recognize the function</strong></p><p>f(x) = (eˣ - e⁻ˣ)/(eˣ + e⁻ˣ) = tanh(x), the hyperbolic tangent function.</p><p><strong>Step 2: Find f'(x) using the quotient rule</strong></p><p>Let u = eˣ - e⁻ˣ, so u' = eˣ + e⁻ˣ</p><p>Let v = eˣ + e⁻ˣ, so v' = eˣ - e⁻ˣ</p><p>f'(x) = (u'v - uv')/v² = [(eˣ + e⁻ˣ)(eˣ + e⁻ˣ) - (eˣ - e⁻ˣ)(eˣ - e⁻ˣ)]/(eˣ + e⁻ˣ)²</p><p><strong>Step 3: Simplify the numerator</strong></p><p>Numerator = (eˣ + e⁻ˣ)² - (eˣ - e⁻ˣ)²</p><p>= (e²ˣ + 2 + e⁻²ˣ) - (e²ˣ - 2 + e⁻²ˣ)</p><p>= e²ˣ + 2 + e⁻²ˣ - e²ˣ + 2 - e⁻²ˣ = 4</p><p><strong>Step 4: Conclude about monotonicity</strong></p><p>f'(x) = 4/(eˣ + e⁻ˣ)² > 0 for all x ∈ ℝ</p><p>Since f'(x) > 0 for all x, f(x) is strictly increasing on ℝ.</p><p><strong>Step 5: Check if f is onto</strong></p><p>As x → ∞, f(x) → 1; as x → -∞, f(x) → -1</p><p>Range of f = (-1, 1) ≠ ℝ, so f is NOT onto.</p><p><strong>∴ Answer: A</strong></p>
Correct Answer: A