Check if the following functions have an inverse function. If yes… - Vidyadip

Question

Check if the following functions have an inverse function. If yes, find the inverse function.

$f(x)=\frac{6 x-7}{3}$

✓

Answer

$f(x)=\frac{6 x-7}{3}$
Let $\mathrm{f}\left(\mathrm{x}_1\right)=\mathrm{f}\left(\mathrm{x}_2\right)$
$\therefore \frac{6 x_1-7}{3}=\frac{6 x_2-7}{3}$
$\therefore \mathrm{x}_1=\mathrm{x}_2$
$\therefore \mathrm{f}$ is a one-one function.
$\mathrm{f}(\mathrm{x})=\frac{6 x-7}{3}=\mathrm{y}$ (say)
$\therefore \mathrm{x}=\frac{3 y+7}{6}$
$\therefore$ For every $\mathrm{y}$, we can get $\mathrm{x}$
$\therefore \mathrm{f}$ is an onto function.
$\therefore \mathrm{x}=\frac{3 y+7}{6}=\mathrm{f}^{-1}(\mathrm{y})$
Replacing y by $x$, we get
$f-1(x)=\frac{3 x+7}{6}$

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