Question

Let $f:R\to R$ be a function defined by $f(x+1)=\frac{f\left(x\right)-5}{f\left(x\right)-3}\forall x\in R$ then which of the following statements are true?

• A. $f\left(2021\right)=f\left(2025\right)$
• B. $f\left(2017\right)=f\left(2020\right)$
• C. $f\left(2018\right)=f\left(2026\right)$
• D. $f\left(7\right)=f\left(10\right)$

Answer 1

Answer: (A), (C)

$f\left(2018\right)=f\left(2026\right)$

$f(x+1)=\frac{f\left(x\right)-5}{f\left(x\right)-3}$
When we replace $x$ with $x+1$ we get
$\Rightarrow f(x+2)=\frac{2f\left(x\right)-5}{f\left(x\right)-2}$
Now replace $x$ with $x+2$ we get
$\Rightarrow f(x+3)=\frac{-3f\left(x\right)+5}{-f\left(x\right)+1}$
Again replace $x$ with $x+1\Rightarrow f(x+4)=f\left(x\right)$
$\therefore f\left(x\right)=f(x+4)\Rightarrow f$ is periodic with period $4.$
So $f\left(2021\right)=f\left(2025\right)$ and $f\left(2018\right)=f\left(2026\right)$ are true.