Maharashtra BoardEnglish MediumSTD 12 ScienceMathsFunctions5 Marks
Question
Give an example of a function:
Which is one-one but not onto.
✓
Answer
Which is one-one but not onto.
f : Z → Z given by f(x) = 3x + 2 Infectivity: Let x andy be any two elements in the domain (Z), such that f(x) = f(y).
f(x) = f(y)
Implies that 3x + 2 = 3y + 2
Implies that 3x = 3y
Implies that x = y
Implies that f(x) = f(y)
Implies that x = y
Therefore,
f is one-one. Subjectivity: Let y be any element in the co-domain (Z), such that f (x) = y for some element x in Z (domain).
f(x) = y
Implies that 3x + 2 = y
Implies that 3x = y - 2
Implies that $\text{x}=\frac{\text{y}-2}{3}.$
It may not be in the domain (Z) because if we take y = 3, $\text{x}=\frac{\text{y}-2}{3}=\frac{3-2}{3}=\frac{1}{3}\notin$ domain Z.
Therefore, for every element in the domain there need not be any element in the domain such that f(x) = y.
Thus, f is not onto.
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