Question
The range of the function $$f\left( x \right) = {}^{7 - x}{P_{x - 3}}$$ is :
A.
$$\left\{ {1,\,2,\,3} \right\}$$
B.
$$\left\{ {1,\,2,\,3,\,4,\,5,\,6} \right\}$$
C.
$$\left\{ {1,\,2,\,3,\,4} \right\}$$
D.
$$\left\{ {1,\,2,\,3,\,4,\,5} \right\}$$
Answer :
$$\left\{ {1,\,2,\,3} \right\}$$
Solution :
The given function $$f\left( x \right) = {}^{7 - x}{P_{x - 3}}$$ would be defined, if
$$\eqalign{
& ({\text{i}})\,7 - x > 0\, \Rightarrow x < 7 \cr
& ({\text{ii}})\,x - 3 \geqslant 0\, \Rightarrow x \geqslant 3 \cr
& ({\text{iii}})\,\left( {x - 3} \right) \leqslant \left( {7 - x} \right) \Rightarrow 2x \leqslant 10\, \Rightarrow x \leqslant 5\, \Rightarrow x = 3,\,4,\,5, \cr
& {\text{Hence, range of }}\,f\left( x \right) = \left\{ {{}^4{P_0},\,{}^3{P_1},\,{}^2{P_2}} \right\} \cr
& {\text{Range of }}\,f\left( x \right) = \left\{ {1,\,3,\,2} \right\} \cr} $$