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Given a function $\frac{4x - 1}{2x + 3}$

I know its Domain is $\{x \in \mathbb{R} | x \ne \frac{-3}{2}\}$

But how can I find its range?

Any hint is appreciated.

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1 Answer 1

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.Write $y = \frac{4x-1}{2x-3}$, and make $x$ the subject of the equation : $$ y = \frac{4x-1}{2x-3} \implies 2xy-3y = 4x- 1 \implies 2xy-4x = 3y-1 \overset{?}{\implies} x = \frac{3y-1}{2y-4} $$

which tells you that for any $y$ such that $\frac{3y-1}{2y-4}$ is in the domain(and such that the last transformation makes sense), we have $f\left(\frac{3y-1}{2y-4}\right) = y$. So any such $y$ is in the range, in fact the range is made up of all such $y$. Now, can you find the range of the function?

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  • $\begingroup$ Thanks for your helpful explanation, now I got some directions to do the similar cases. $\endgroup$ Mar 21, 2019 at 6:25
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    $\begingroup$ You are welcome! Also note that you can close the question if you are satisfied : you can refer to it later if you are having similar questions. $\endgroup$ Mar 21, 2019 at 6:26

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