Sets, Relations & Functions
General
Grade 11
Question:
<p>Let <span class="math-inline">\(f: \mathbb{R}\setminus\{-\frac{15}{2}\} \to \mathbb{R}\setminus\{\frac{1}{2}\}\)</span>, <span class="math-inline">\(f(x) = \dfrac{x+10}{2x+15}\)</span>. Determine whether <span class="math-inline">\(f\)</span> 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
Key Concept: General
<div class="solution"><p><strong>Key Idea:</strong> For a Möbius (linear fractional) function, solve for <span class="math-inline">$x$</span> in terms of <span class="math-inline">$y$</span> directly.</p><p><strong>Step 1:</strong> <span class="math-block">$$y = \frac{x+10}{2x+15} \implies x = \frac{10-15y}{2y-1}$$</span>This exists for all <span class="math-inline">$y \ne \frac{1}{2}$</span> — 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><div class="key-concept"><strong>Key Concept:</strong> Möbius functions — bijectivity via explicit inverse</div></div>
Correct Answer: 3