Sets, Relations & Functions
General
Grade 11
Question:
<p>Let <span class="math-inline">\(f(x)=\dfrac{x-2}{x-3}\)</span>, <span class="math-inline">\(g(x)=2x-3\)</span>. Find sum of all <span class="math-inline">\(x\)</span> where <span class="math-inline">\(f^{-1}(x)+g^{-1}(x)=\frac{13}{2}\)</span>.</p>
Step-by-Step Solution
Key Concept: General
<div class="solution"><p>f⁻¹(x)=(3x-2)/(x-1); g⁻¹(x)=(x+3)/2. Sum=(3x-2)/(x-1)+(x+3)/2=13/2. Solving: multiply out, get quadratic with sum of roots=5.</p><div class="key-concept"><strong>Key Concept:</strong> Explicit inverse + Vieta's sum of roots</div></div>
Correct Answer: 5