Limits, Continuity & Differentiability
General
Grade 12
Question:
<p><span class="math-inline">\(f:\mathbb{R}\to\mathbb{R}\)</span>, <span class="math-inline">\(g(x)=|f(x)|\)</span>. Which are NOT always true?</p>
f cont⟹g cont
<strong>f one-one⟹g one-one</strong>
<strong>f onto⟹g onto</strong>
<strong>f diff⟹g diff</strong>
Step-by-Step Solution
Key Concept: General
<div class="solution"><p>(A) f continuous ⟹ g continuous: TRUE always (composition of continuous functions).</p><p>(B) f one-one ⟹ g one-one: FALSE. e.g., f(x)=x, g(x)=|x| — g is not one-one. NOT always true. ✓</p><p>(C) f onto ⟹ g onto: FALSE. If f is onto ℝ, g=|f| maps to [0,∞), not all of ℝ. NOT always true. ✓</p><p>(D) f differentiable ⟹ g differentiable: FALSE. f(x)=x, g(x)=|x| not differentiable at 0. NOT always true. ✓</p><p><strong>Answer: (B),(C),(D)</strong></p><div class="key-concept"><strong>Key Concept:</strong> Absolute value destroys injectivity, surjectivity, and differentiability at zeros of f</div></div>
Correct Answer: B,C,D