Question

determine whether function f(x) = 2x+1/x-2 has inverse, if it exists find it.​

Answer 1

Answer:

Step-by-step explanation:

Hopefully this will help,

Answer 2

SOLUTION

TO CHECK

Whether inverse of the below function exists :

$\displaystyle \sf{f(x) = \frac{2x + 1}{x - 2} }$

CONCEPT TO BE IMPLEMENTED

For a given real valued function the inverse of the function does not exist at the point where it vanishes

EVALUATION

Here the given function is

$\displaystyle \sf{f(x) = \frac{2x + 1}{x - 2} }$

Now f(x) does not exists at x = 2

So the inverse does not exist at x = 2

Now suppose domain = R - { 2 }

$\sf{Let \: \: \: {f}^{ - 1}(x) = y }$

$\sf{ \implies \: x = f(y)}$

$\displaystyle \sf{ \implies \: x = \frac{2y + 1}{y - 2} }$

$\displaystyle \sf{ \implies \: xy - 2x = 2y+ 1 }$

$\displaystyle \sf{ \implies \: xy - 2y = 2x+ 1 }$

$\displaystyle \sf{ \implies \: y(x- 2) = 2x + 1 }$

$\displaystyle \sf{ \implies \: y= \frac{2x+ 1}{x - 2} }$

$\displaystyle \sf{ \implies \: {f}^{ - 1} (x)= \frac{2x+ 1}{x - 2} }$

FINAL ANSWER

$\displaystyle \sf{ \: {f}^{ - 1} (x)= \frac{2x+ 1}{x - 2} } \: \: \: where \: \: x \in \: \mathbb{R} - \{ 2\}$

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