# 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}

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