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Functions — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsFunctions5 Marks
Question
Classify the following functions as injection, surjection or bijection:
f : Q - {3} → Q, defined by $\text{f(x)}=\frac{2\text{x}+3}{\text{x}-3}$
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Answer

f : Q - {3} → Q, defined by $\text{f(x)}=\frac{2\text{x}+3}{\text{x}-3}$
Injection test: Let x and y be any two elements in the domain (Q - {3}), such that f(x) = f(y).
f(x) = f(y)
$\frac{2\text{x}+3}{\text{x}-3}=\frac{2\text{y}+3}{\text{y}-3}$
(2x + 3)(y - 3) = (2y + 3)(x - 3)
2xy - 6x + 3y - 9 = 2xy -6y + 3x - 9
9x = 9y
x = y
Therefore, f is an injection.
Surjection test: Let y be any element in the co-domain (Q - {3}), such that f(x) = y for some element x in Q (domain).
f(x) = y
$\frac{2\text{x}+3}{\text{x}-3}=\text{y}$
2x + 3 = xy - 3y
2x - xy = -3y - 3
x(2 - y) = -3(y + 1)
$\text{x}=\frac{3\text{y}+1}{\text{y}-2},$ which is not defined at y = 2.
Therefore, f is not a surjection and f is not a bijection.

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