Sets, Relations & Functions
Functional Equation — Solving for f(5)−f(2)
nta_pyq_2026_jan
Grade 11
Question:
Let $f$ be a function such that $3f(x)+2f\!\left(\dfrac{m}{19x}\right)=5x$, $x\neq0$, where $m=\displaystyle\sum_{i=1}^9 i^2$. Then $f(5)-f(2)$ is equal to
Step-by-Step Solution
Key Concept: $m=\tfrac{9\cdot10\cdot19}{6}=285$, so $\tfrac{m}{19}=15$. Equation: $3f(x)+2f\!\left(\tfrac{15}{x}\right)=5x$ ...(i). Replace $x$ by $\tfrac{15}{x}$: $3f\!\left(\tfrac{15}{x}\right)+2f(x)=\tfrac{75}{x}$ ...(ii).
$f(x)=3x-\tfrac{30}{x}$. $f(5)-f(2)=9-(-9)=18$.
Correct Answer: 1