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Question

Let A =R -{3}, B=R -{1}. If f:AB be defined by f(x)=x2x3,xA. Then, show that f is bijective.


Solution

Given that, A=R -{3}, B=R -{1}
f:AB is defined by f(x)=x2x3,xA
For injectivity
Let f(x1)=f(x2)x12x13=x22x23
(x12)(x23)=(x22)(x13)x1x23x12x2+6=x1x23x22x1+63x12x2=3x22x1x1=x2x1=x2
So, f(x)is an injective function.

For surjectivity
Let y=x2x3x2=xy3y
x(1y)=23yx=23y1y
x=3y2y1A,yB [codomain]
So, f(x)is surjective function.

Hence, f(x) is a bijective function


Mathematics

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