Let f : R - n → R be a function defined by f(x)= x-m x-n , where … - Vidyadip

MCQ

Let f : R - {n} → R be a function defined by $\text{f(x)}=\frac{\text{x}-\text{m}}{\text{x}-\text{n}},$ where $\text{m}\neq\text{n.}$ Then,

A
f is one-one onto.
✓
f is one-one into.
C
f is many one onto.
D
f is many one into.

✓

Answer

Correct option: B.

f is one-one into.

Given function f : R - {n} → R be a function defined by $\text{f(x)}=\frac{\text{x}-\text{m}}{\text{x}-\text{n}},\ \text{m}\neq\text{n}$

If f(x) = f(y) then

$\frac{\text{x}-\text{m}}{\text{x}-\text{n}}=\frac{\text{y}-\text{m}}{\text{y}-\text{n}}$

⇒ (x - m)(y - n) = (y - m)(x - n)

After solving this we will get x = y

Hence, it is one-one.

$\text{f(x)}=\frac{\text{x}-\text{m}}{\text{x}-\text{n}},\ \text{m}\neq\text{n}$

$\text{y}=\frac{\text{x}-\text{m}}{\text{x}-\text{n}}$

⇒ y(x - n) = x - m

⇒ yx - yn = x - m

⇒ yx - x = ny - m

⇒ x(y - 1) = ny - m

$\Rightarrow\ \text{x}=\frac{\text{ny}-\text{m}}{\text{y}-1}$

Here, for y = 1 we can not define x.

Hence, it is not onto.

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