If (f:{0,1,2,3,4}\to{0,1,2,3,4}) is given by (f(x)=x^2-3x+2), is (f) a valid function?

Correct answer: No, because (f(4)=6\notin{0,1,2,3,4})

The direct answer is B: no, this is not a valid function from the stated domain to the stated codomain. A function must assign every input exactly one output, and that output must belong to the codomain. Check the inputs one by one: f(0)=0^2−3(0)+2=2, f(1)=1−3+2=0, f(2)=4−6+2=0, f(3)=9−9+2=2, and f(4)=16−12+2=6. Although each input has one value, 6 is not in {0,1,2,3,4}; therefore the mapping fails as a function into the given codomain. Option A is wrong because not all values lie in the codomain. Option B is correct because f(4)=6 is outside it. Option C is wrong because f(1)=0 is allowed and does not cause a problem. Option D is wrong: a polynomial can be evaluated on a finite set, but its outputs must still lie in the declared codomain. Memory cue: check every input and then check whether every output belongs to the codomain.