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

For the function $f(x)={}^{7-x}{P}_{x-3}$ to be defined, the following conditions for ${}^n{P}_r$ must be satisfied:$n\ge 0\Rightarrow 7-x\ge 0\Rightarrow x\le 7$.$r\ge 0\Rightarrow x-3\ge 0\Rightarrow x\ge 3$.$n\ge r\Rightarrow 7-x\ge x-3\Rightarrow 10\ge 2x\Rightarrow x\le 5$.Combining these, we get $3\le x\le 5$. Since $x$ must be an integer for the permutation formula to hold in this context, $x\in \{3,4,5\}$.
Now, calculate the function values:For $x=3:f(3)={}^{7-3}{P}_{3-3}={}^4{P}_0=1$.For $x=4:f(4)={}^{7-4}{P}_{4-3}={}^3{P}_1=3$.For $x=5:f(5)={}^{7-5}{P}_{5-3}={}^2{P}_2=2$.The range of the function is $\{1,2,3\}$.