The function f is mapped from natural numbers to integer numbers and f(x) = x2…
The function f is mapped from natural numbers to integer numbers and f(x) = x2 - 3x + 3 then what is the function f?
Answer: D. None of the above — Given f: N → Z defined by f(x) = x^2 - 3x + 3. Show not injective: Compute f(1) = 1^2 - 3·1 + 3 = 1 and f(2) = 2^2 - 3·2 + 3 = 1. Two distinct inputs (1 and…
- A.
Injective function
- B.
Surjective function
- C.
Bijective function
- D.
None of the above
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Show answer & explanation
Correct answer: D
Given f: N → Z defined by f(x) = x^2 - 3x + 3.
Show not injective:
Compute f(1) = 1^2 - 3·1 + 3 = 1 and f(2) = 2^2 - 3·2 + 3 = 1.
Two distinct inputs (1 and 2) give the same output, so the function is not injective.
Show not surjective:
Check whether 0 is in the image: solve x^2 - 3x + 3 = 0.
The discriminant is Δ = (-3)^2 - 4·1·3 = 9 - 12 = -3, which is negative, so there are no real solutions and hence no natural solutions for f(x) = 0.
Therefore 0 is not in the image of f, so f is not surjective onto Z.
Conclusion:
Since f is neither injective nor surjective, it is not bijective.
Therefore the correct choice is that none of the listed properties (injective, surjective, bijective) hold.