Sets, Relations & Functions
Functional Equations
nta_pyq_2023_jan
Grade 11
Question:
Let $f: \mathbb{R} - \{0,1\} \to \mathbb{R}$ be a function such that $f(x) + f\!\left(\dfrac{1}{1-x}\right) = 1 + x$. Then $f(2)$ is equal to:
$\dfrac{9}{2}$
$\dfrac{9}{4}$
$\dfrac{7}{4}$
$\dfrac{7}{3}$
Step-by-Step Solution
Key Concept: Substitute $x=2$, $x=\frac{1}{1-2}=-1$, $x=\frac{1}{1-(-1)}=\frac{1}{2}$ to form 3 equations and solve for $f(2)$.
Eq1: $f(2)+f(-1)=3$. Eq2: $f(-1)+f(1/2)=0$. Eq3: $f(1/2)+f(2)=5/2$. Solving: $f(2)=9/4$.
Correct Answer: 2