Question

The range of the function $f\left(x\right){=}^{7-x}{P}_{x-3}$ is
(a) {1, 2, 3, 4, 5}
(b) {1, 2, 3, 4, 5, 6}
(c) {1, 2, 3, 4}
(d) {1, 2, 3}

Answer 1

We know that
$$\begin{gathered} 7-x>0;x-3\ge 0\text{and}7-x\ge x-3 \\ \Rightarrow x<7;x\ge 3\text{and}2x\le 10 \\ \Rightarrow x<7;x\ge 3\text{and}x\le 5 \\ \text{So,}x=\left\{3,4,5\right\} \\ \text{Range of}\mathit{\text{f}} \\ \mathit{\text{=}}\left\{{}^{\left(\mathit{7}\mathit{-}\mathit{3}\right)}P_{\left(\mathit{3}\mathit{-}\mathit{3}\right)}\mathit{,}\mathit{}{}^{\left(\mathit{7}\mathit{-}\mathit{4}\mathit{}\right)}P_{\left(\mathit{4}\mathit{-}\mathit{3}\right)}\mathit{,}\mathit{}{}^{\left(\mathit{7}\mathit{-}\mathit{5}\right)}P\left({}_{\mathit{5}\mathit{-}\mathit{3}}\right)\right\} \\ \mathit{\text{=}}\left\{\mathit{4}{P}_{\mathit{0}}\mathit{,}\mathit{}\mathit{3}{P}_{\mathit{1}}\mathit{,}\mathit{}\mathit{2}{P}_{\mathit{2}}\right\} \\ \mathit{\text{=}}\left\{\mathit{1}\mathit{,}\mathit{}\mathit{3}\mathit{,}\mathit{}\mathit{2}\right\} \\ \mathit{\text{=}}\left\{\mathit{1}\mathit{,}\mathit{}\mathit{2}\mathit{,}\mathit{}\mathit{3}\right\} \end{gathered}$$
So, the answer is (d).