Question

The range of the function $f\left(x\right)={}^{7-x}{P}_{x-3}$ is:

• A. $\{1,2,3\}$
• B. $\{1,2,3,4,5\}$
• C. $\{1,2,3,4\}$
• D. $\{1,2,3,4,5,6\}$

Answer 1

Answer: (A)

$\{1,2,3\}$

$f\left(x\right){=}^{7-x}{P}_{x-3}$

$\therefore$ $f\left(x\right)$ is defined when $x-3\ge 0$

$\Rightarrow x\ge 3$ ....(1)

Again $f\left(x\right){=}^{7-x}{P}_{x-3}=\frac{(7-x)!}{(10-2x)!}$

$\therefore$ $f\left(x\right)$ is defined when $(10-2x)\ge 0$

$\Rightarrow 2x\le 10$

$\Rightarrow x\le 5$ ....(2)

From (1) and (2), we have

$3\le x\le 5$
$\therefore$ Range $\left\{f\right(3),f(4),f(5\left)\right\}$

$=\left\{{}^4{P}_1{,}^3{P}_1{,}^2{P}_2\right\}$

$=\{1,3,2\}$