First slide

Functions (XII)

Question

Let $f : R - {n} \to R$ be a function defined by $f (x) = \frac{x - m}{x - n}$ such that $m \neq n$ then

Easy

A

f is one one into function
B

f is one one onto function
C

f is many one into function
D

f is many one onto function

Solution

Given $f (x) = \frac{x - m}{x - n}$
where $m \neq n, \forall x \in R - {n}, Let x_{1}, x_{2} \in R$
$∴ f (x_{1}) = f (x_{2}) \Rightarrow \frac{x_{1} - m}{x_{1} - n} = \frac{x_{2} - m}{x_{2} - n} \Rightarrow x_{1} = x_{2}$
$∴ f$ is one-one
Let $λ \in R$ such that $f (x) = λ, ∴ \frac{x - m}{x - n} = λ$
$∴ x = \frac{m - n λ}{1 - λ}$
x is not defined for $λ = 1,$
$∴ f (x)$ is not onto function.
If a function is not onto it refered that it is into function. Hence f is one-one into function.

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