Let A=R−{3} and B=R−{1} Let f:A→B:f...
Question
Let A=R−{3} and B=R−{1} Let f:A→B:f(x)=x−2/x−3 for all values of x∈A Show that fis one-one and onto.
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Let ( A=R-{3} ) and ( B=R-{1} ) Let ( f: A rightarrow B: f(x)=frac{x-2}{x-3} ) for all values of ( x in A ) Show that fis one-one and onto.

11th - 12th Class
Maths
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Let A=R−{3} and B=R−{1} Let f:A→B:f(x)=x−2/x−3 for all values of x∈A Show that fis one-one and onto.
Fullscreen

( quad f ) is one-one, since
[
begin{aligned}
fleft(x_{1}right)=fleft(x_{2}right) & Rightarrow frac{x_{1}-2}{x_{1}-3}=frac{x_{2}-2}{x_{2}-3}
& Rightarrowleft(x_{1}-2right)left(x_{2}-3right)=left(x_{1}-3right)left(x_{2}-2right)
& Rightarrow x_{1} x_{2}-3 x_{1}-2 x_{2}+6=x_{1} x_{2}-2 x_{1}-3 x_{2}+6
& Rightarrow x_{1}=x_{2}
end{aligned}
]
Let ( y in B ) such that ( y=frac{x-2}{x-3} )
Then, ( (x-3) y=(x-2) Rightarrow x=frac{(3 y-2)}{(y-1)} )
Clearly, ( x ) is defined when ( y eq 1 ) Also, ( x=3 ) will give us ( 1=0, ) which is false.
( therefore quad x eq 3 )
[
text { And, } f(x)=frac{left(frac{3 y-2}{y-1}-2right)}{left(frac{3 y-2}{y-1}-3right)}=y
]
Thus, for each ( y in B ), there exists ( x in A ) such that ( f(x)=y ).
( therefore f ) is onto.
Hence, ( f ) is one-one onto.

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