Sets, Relations & Functions
Grade 11
Question:
<p>Let \(f: \mathbb{R}\setminus\{-\frac{15}{2}\} \to \mathbb{R}\setminus\{\frac{1}{2}\}\), \(f(x) = \dfrac{x+10}{2x+15}\). Determine whether \(f\) is one-one and/or onto.</p>
One-one but not onto
Onto but not one-one
<strong>One-one and onto</strong>
Neither one-one nor onto
Step-by-Step Solution
<div class="solution"><p><strong>Key Idea:</strong> For a Möbius (linear fractional) function, solve for \(x\) in terms of \(y\) directly.</p><p><strong>Step 1:</strong> <span class="math-block">\[y = \frac{x+10}{2x+15} \implies x = \frac{10-15y}{2y-1}\]This exists for all \(y \ne \frac{1}{2}\) -- so every element of the codomain has exactly one preimage.</p><p><strong>Answer: One-one and onto</strong></p><div class="trap-box"><strong>Trap:</strong> Do not test injectivity by graph sketch. The explicit inverse settles both properties in one step.<div class="key-concept"><strong>Key Concept:</strong> Möbius functions -- bijectivity via explicit inverse
Correct Answer: 3