Question

Let $Z$ be the set of integers and $f:Z\to Z$ is a bijective function then

• A. $f\left(x\right)=x^2+4$
• B. $f\left(x\right)=2x+3$
• C. $f\left(x\right)=x$
• D. $f\left(x\right)=x^3+3$

Answer 1

The correct option is C $f\left(x\right)=x$

Let us check all the options one by one:

Option A:

$f\left(x\right)=x^2+4$
Here, $f\left(1\right)=1^2+4=1+4=5$

$f(-1)=(-1{)}^2+4=1+4=5$

Thus, $f\left(1\right)=f(-1)$but $1\ne -1$

So, it is not one - one and hence not a bijective function.

Option B:
$f\left(x\right)=2x+3$
It is one - one but it is not onto, because for $y=0,x=\frac{-3}{2}\notin Z$

Hence, not a bijective function.

Option 3:
$f\left(x\right)=x$
It is one - one & onto. So it is a bijective function.

$f\left(x\right)=x^3+3$
it is one - one but not onto, because for $y=0,x\notin Z$

Hence, not a bijective function.