Let A = R - 3 and B = R - 1 . Consider the function f : A B defin… - Vidyadip

Question

Let $A = R - {3}$ and $B = R - {1}$. Consider the function $f : A \rightarrow B$ defined by $\text{f(x)}=\frac{\text{x}-2}{\text{x}-3}.$ Show that f is one-one and onto and hence find $f^{-1}.$

✓

Answer

We have,
$A = R - {3}$ and $B = R - {1}$. Consider the function $f : A \rightarrow B$ defined by
$\text{f}(\text{x})=\frac{\text{x}-2}{\text{x}-3},$ Show that f is one-one and onto and hence find $f^{-1}.$
Let $\text{x, y}\in\text{A}$ such that f(x) = f(y). Then,
$\frac{\text{x}-2}{\text{x}-3}=\frac{\text{y}-2}{\text{y}-3}$
$\Rightarrow xy - 3x - 2y + 6 = xy - 2x - 3y + 6$
$\Rightarrow -x = -y$
$\Rightarrow x = y$
$\therefore$ f is one-one.

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