Question

If $f\left(x\right){=}^{7-x}{P}_{x-3}$ find domain and range of $f\left(x\right)$.

• A. range is ${}^4{P}_0{,}^3{P}_1{,}^2{P}_2$ and domain is $\{3,4,5\}$
• B. domain is ${}^4{P}_0{,}^3{P}_1{,}^2{P}_2{,}^1{P}_1$ and range is $\{2,3,4,5\}$
• C. range is ${}^4{P}_0{,}^3{P}_1{,}^2{P}_2{,}^1{P}_1$ and domain is $\{1,2,3,4\}$
• D. none of these

Answer 1

The correct option is A range is ${}^4{P}_0{,}^3{P}_1{,}^2{P}_2$ and domain is $\{3,4,5\}$

${}^n{P}_r$ is defined if both n and r are+ive integers and $r\le n$.

$\therefore x\le 7,x\ge 3andx-3\le 7-xi.e.x\le 5$

all the three conditions imply and we get, $3\le x\le 5$

$\therefore$ Domain consists of integers $\{3,4,5\}$

Range$=y$ for $x=3,4,5{,}^{7-x}{P}_{x-3}$ i.e ${}^4{P}_0{,}^3{P}_1{,}^2{P}_{2}or\{1,2,3\}$