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Questions  >>  CBSE XII  >>  Math  >>  Relations and Functions
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Q)

Which of the following functions from Z into Z are bijections?\begin{array}{1 1}(A)\;f(x)=x^3 & (B)\;f(x)=x+2\\(C)\;f(x)=2x+1 & (D)\;f(x)=x^2+1\end{array}

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A)
Toolbox:
  • A function $f: Z \to Z$ is bijective if f is both one -one and onto
  • ie $f(x)=f(y) =>x =y$
  • and for every $y \in R$ then exists $ x\in R $ such that $f(x)=y$
$f(x)=x^3 \qquad x \in z$
 
$f(x_1)=f(x_2)$
 
$x_1^3=x_2^3$
 
$x_1 =x_2$
 
f is one one
 
But for $y=-2$ then does not exists $x \in Z$ such that $f(x)=-2$ ie $x^3=-2$
 
f is not onto
 
f is not bijection
 
$f(x)=x+2$
 
$f(x_1)=f(x_2)$
 
$=> x_1+2=x_2+2$
 
$x_1=x_2$
 
f is one one
 
Also $y=x+2 \qquad \in z$ then there exists
 
$x=y-2 \qquad \in z$ such that
 
$f(x)=y$
 
$f(y-z)=y-z+z$
 
$=y$
 
f is onto
 
Hence f=x+2 is bijection
 
'B' option is correct

 

 

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