Question

If a function $f:\{1,2,3,4\}\to \{1,2,3,4,5,6,7,8,9\}$ is defined, then the function $f$ can be

• A. $f\left(x\right)=2x+1$
• B. $f\left(x\right)=|x|$
• C. $f\left(x\right)=x^2$
• D. $f\left(x\right)=x-1$

Answer 1

Answer: (A), (B)

$f\left(x\right)=|x|$

Since function is defined as $f:\{1,2,3,4\}\to \{1,2,3,4,5,6,7,8,9\}$
$D\left(f\right)=\{1,2,3,4\}$
Here, each and every element of $D\left(f\right)$ should be in $\{1,2,3,4,5,6,7,8,9\}$

$f\left(x\right)=2x+1$
$f\left(1\right)=2\left(1\right)+1=3$
$f\left(2\right)=5,f\left(3\right)=7,f\left(4\right)=9$
Here, $f\left(1\right),f\left(2\right),f\left(3\right),f\left(4\right)\in \{1,2,3,4,5,6,7,8,9\}$
So, it can be the required function.

Similarly, $f\left(x\right)=|x|$
$f\left(1\right)=1,f\left(2\right)=2,f\left(3\right)=3,f\left(4\right)=4$
$f\left(1\right),f\left(2\right),f\left(3\right),f\left(4\right)\in \{1,2,3,4,5,6,7,8,9\}$
So, it can be the required function.

$f\left(x\right)=x^2$
$f\left(1\right)=1,f\left(2\right)=4,f\left(3\right)=9,f\left(4\right)=16$
$f\left(1\right),f\left(2\right),f\left(3\right)\in \{1,2,3,4,5,6,7,8,9\}$
But $f\left(4\right)\notin \{1,2,3,4,5,6,7,8,9\}$
So, it cannot be the required function.

$f\left(x\right)=x-1$
$f\left(1\right)=0,f\left(2\right)=1,f\left(3\right)=2,f\left(4\right)=3$
$f\left(2\right),f\left(3\right),f\left(4\right)\in \{1,2,3,4,5,6,7,8,9\}$
But $f\left(1\right)\notin \{1,2,3,4,5,6,7,8,9\}$
So, it cannot be the required function.