Question

IF f(x)=x÷(x-1)
find(1) f o f(x)
(2)inverse of f​

Answer 1

$\begin{gathered}\begin{gathered}\bf Given -  \begin{cases} &\sf{f(x) = \dfrac{x}{x - 1} } \end{cases}\end{gathered}\end{gathered}$

$\begin{gathered}\begin{gathered}\bf  To \:  Find :-  \begin{cases} &\sf{f \: o \: f \: (x)} \\ &\sf{ {f}^{ - 1}(x) } \end{cases}\end{gathered}\end{gathered}$

$\large\underline\purple{\bold{Solution :- }}$

$\tt \ \: : Given \: f(x) \: = \dfrac{x}{x - 1}$

$\displaystyle\longrightarrow \: \tt \: Now \: \: \: fof(x) \: = \: f \: \{f(x) \}$

$\displaystyle\longrightarrow \tt \: = f \bigg(\dfrac{x}{x - 1} \bigg)$

$\displaystyle\longrightarrow \tt \: = \dfrac{\dfrac{x}{x - 1}}{\dfrac{x}{x - 1} - 1}$

$\displaystyle\longrightarrow \tt \: = \dfrac{\dfrac{x}{x - 1}}{\dfrac{ \cancel{x} - \cancel{x} + 1}{x - 1}}$

$\displaystyle\longrightarrow \tt \: = \dfrac{x}{x - 1} \times (x - 1)$

$\bf\implies \boxed{\:fof(x) \: = x}$

$\bf\implies \:f(x) \: = \: {f}^{ - 1} (x)$

$\bf\implies \boxed{\: {f}^{ - 1} (x) = \dfrac{x}{x - 1}}$