Maharashtra BoardEnglish MediumSTD 12 ScienceMathsFunctions5 Marks
Question
Give an example of a function:
Which is neither one-one nor onto.
✓
Answer
Which is neither one-one nor onto.
$f : Z \rightarrow Z$ given by $f(x) = 2x^2 + 1$ Infectivity: Let x andy be any two elements in the domain $(Z),$ such that $f(x) = f(y).$
$f(x) = f(y)$
Implies that $2x^2 + 1 = 2y^2 + 1$
Implies that $2x^2 = 2y^2$
Implies that $x^2 = y^2$
Implies that $\text{x}=\pm\text{y}$
Therefore, different elements of domain/ may give the same image.
Thus, $f$ is not 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 $2x^2 + 1 = y$
Implies that $2x^2 = y - 1$
Implies that $\text{x}^2=\frac{\text{y}-1}{2}$
Implies that $\text{x}=\pm\sqrt{\frac{\text{y}-1}{2}},\notin\text{Z}$ always.
For example, if we take, $y = 4,$
$\text{x}=\pm\sqrt{\frac{\text{y}-1}{2}}=\pm\sqrt{\frac{4-1}{2}}=\pm\sqrt{\frac{3}{2}},\notin\text{Z}$
Therefore, $x$ may not be in $Z ($domain$).$
Thus, $f$ is not onto.
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