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Q.
If the function $f (x) = ax + b$ has its own inverse then the ordered pair$ (a, b) $ can be
Relations and Functions - Part 2
Solution:
$ y =f( x ) \Rightarrow x =f^{-1}( x )$
$\text { now } y = ax + b$
$x =\frac{ y }{ a }-\frac{ b }{ a }$
$f^{-1}( y )=\frac{ y }{ a }-\frac{ b }{ a }$
$f^{-1}( x )=\frac{ x }{ a }-\frac{ b }{ a }$....(1)
$\text { and } f ( x )= ax + b \ldots .(2) $
$\text { now in order that (1) and (2) coincide } $
$a =\frac{1}{ a } $....(1)
$\frac{ b }{ a }=- b$....(2)
$\text { from (1), } a ^2=1 \Rightarrow a =1 \text { or }-1 $
$\text { if } a =-1, b = b \Rightarrow b \in R $
$\text { if } a=+1 \text {, then } 2 b=0 \Rightarrow b=0$
$\text { hence }(-1, R ),(1,0)$