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The range of the function f(x) = $${}^{7 - x}{P_{x - 3}}$$ is

JEE · Math · previous-year question

  1. A.{1, 2, 3, 4, 5}
  2. B.{1, 2, 3, 4, 5, 6}
  3. C.{1, 2, 3, 4}
  4. D.{1, 2, 3}correct

Answer

D. {1, 2, 3}

Explanation

The range of the function $f(x) = {}^{7-x}P_{x-3}$ can be found by considering the possible values of $f(x)$ as $x$ varies over its domain. The domain of $f(x)$ is the set of all real numbers such that (i) $x \geq 3$ (since the permutation function is only defined for non-negative integers) (ii) 7 - x > 0 $$ \Rightarrow $$ x < 7 (iii) x - 3 $$ \le $$ 7 - x $$ \Rightarrow $$ 2x $$ \le $$ 10 $$ \Rightarrow $$ x $$ \le $$ 5. To find the range of $f(x)$, we need to consider what values the expression ${}^{7-x}P_{x-3}$ can take as $x$ varies over its domain. For $x=3$, we have ${}^{7-x}P_{x-3} = {}^{4}P_{0} = 1$. For $x=4$, we have ${}^{7-x}P_{x-3} = {}^{3}P_{1} = 3$. For $x=5$, we have ${}^{7-x}P_{x-3} = {}^{2}P_{2} = 2$. Therefore, the range of $f(x)$ is the set of all non-negative integers less than or equal to 3, i.e., ${1, 2, 3}$.

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