JEE Main 2004 — Functions Question with Solution

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) $x\ge 3$ (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., $1,2,3$.