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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}

1 Answer

  • 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 $f(x)=y$
$f(x)=x^3 \qquad x \in z$
$x_1 =x_2$
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
$=> x_1+2=x_2+2$
f is one one
Also $y=x+2 \qquad \in z$ then there exists
$x=y-2 \qquad \in z$ such that
f is onto
Hence f=x+2 is bijection
'B' option is correct



answered Mar 5, 2013 by meena.p

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