Functions Question 14
Question 14 - 30 January - Shift 1
Let $f^{1}(x)=\frac{3 x+2}{2 x+3}, x \in R-{\frac{-3}{2}}$
For $n \geq 2$, define $f^{n}(x)=f^{1}of^{n-1}(x)$.
If $f^{5}(x)=\frac{ax+b}{bx+a}, gcd(a, b)=1$, then $a+b$ is equal to _________
Show Answer
Answer: 3125
Solution:
Formula: Composition of functions, Operations on functions
$ \begin{aligned} & f^{1}(x)=\frac{3 x+2}{2 x+3} \\ & \Rightarrow f^{2}(x)=\frac{13 x+12}{12 x+13} \\ & \Rightarrow f^{3}(x)=\frac{63 x+62}{62 x+63} \\ & \therefore f^{5}(x)=\frac{1563 x+1562}{1562 x+1563} \\ & a+b=3125 \end{aligned} $
