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Show that $f:[-1,1] \rightarrow \mathbf{R}$, given by $f(x)=\frac{x}{(x+2)}$ is one-one. Find the inverse of the function $f:[-1,1] \rightarrow$ Range $f$.
written 3.6 years ago by gravatar for prajapatijaimin prajapatijaimin • 3.3k •   modified 3.6 years ago
discrete structures and graph theory
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written 3.6 years ago by gravatar for prajapatijaimin prajapatijaimin • 3.3k •   modified 3.6 years ago

Solution:

$ f:[-1,1] \rightarrow R \\ $ is given as,

$$ f(x)=\frac{x}{(x+2)} \\ $$

Let, $$ f(x)=f(y) \\ $$

$$ \begin{aligned} &\Rightarrow \frac{x}{x+2}=\frac{y}{y+2} \\\\ &\Rightarrow x y+2 x=x y+2 y \\\\ &\Rightarrow 2 x=2 y \\\\ &\Rightarrow x=y \\ \end{aligned} $$

$\therefore f$, is a one-one function.

It is clear …

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