Let the function f : R - -b → R - 1 be defined by f(x)= x+a x+b ,… - Vidyadip

MCQ

Let the function f : R - {-b} → R - {1} be defined by $\text{f(x)}=\frac{\text{x}+\text{a}}{\text{x}+\text{b}},\ \text{a}\neq\text{b}.$ Then,

A
f is one-one but not onto.
B
f is onto but not one-one.
✓
f is both one-one and onto.
D
None of these.

✓

Answer

Correct option: C.

f is both one-one and onto.

Injectivity: Let x and y be two elements in the domain R - {-b}, such that

f(x) = f(y) ⇒ x + ax + b = y + ay + b

⇒ x + ay + b = x + by + a

⇒ xy + bx + ay + ab = xy + ax + by + ab

⇒ bx + ay = ax + by

⇒ a - bx = a - by

⇒ x = y

So, f is one-one.

Surjectivity: Let y be an element in the co-domain of f,

i.e., R - {1}, such that f(x) = y

⇒ x + ax + b = y

⇒ x + a ⇒ x = -a

So, f is onto.

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