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Let the function f: R-{-b}->R-{1} be def...

Let the function `f: R-{-b}->R-{1}` be defined by `f(x)=(x+a)/(x+b)` , `a!=b` , then `f` is one-one but not onto (b) `f` is onto but not one-one (c) `f` is both one-one and onto (d) none of these

A

f is one-one but not onto

B

f is onto but not one-one

C

f is both one-one and onto

D

None of these

Text Solution

Verified by Experts

The correct Answer is:
C
`Rightarrow (x+a)/(x+b)=(y+a)/(y+b) Rightarrow 1+(a-b)/(x+b)=1+(a-b)/(a+b) Rightarrow x=y`
So, if one-one
Let `y in R` such that f(x)=y. Then.
`f(x)=y Rightarrow (x+a)/(x+b)=y Rightarrow x=(a-by)/(y-1)`
`"Clearly", x in R -(-b)"for all "y in R -(-1).So,
Hence, f is both one-one and onto.
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