Functions
General
Grade 12
Question:
The function f : [0,3] → [1,29], defined by f(x) = \(2x^3 - 15x^2 + 36x + 1\), is :
one-one and onto
onto but not one-one
one-one but not onto
neither one-one nor onto
Step-by-Step Solution
Key Concept: General
<div class="solution"><p>Inner log is odd (using (sec x+tan x)(sec x-tan x)=1). Cubing preserves oddness → f odd. Strictly increasing → one-one. Range is all ℝ → onto.</p><p><strong>Answer: (A),(B),(C)</strong></p><div class="trap-box"><strong>Trap:</strong> The reciprocal identity immediately reveals the odd nature of log(sec x+tan x).</div><div class="key-concept"><strong>Key Concept:</strong> Odd + strictly monotone + limits ±∞ → bijective onto ℝ</div></div>
Correct Answer: B