Question 236724
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I think what you are trying to say is:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ y\ =\ \frac{5x}{3x\ +\ 9}]


and that you want to describe any restrictions on the <b><i>domain</i></b> (not the "remain") of the function.


The domain of any rational function must exclude any value of the independent variable (the *[tex \Large x] in this case) that would make any denominator in the function equal zero.  You have but one denominator here, so let's set it equal to zero and solve:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ 3x\ + 9\ = 0]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ 3x\ =\ - 9]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ x\ =\ -3]


So now we know that -3 is the value that must be excluded from the domain.


All other real values for the independent variable are acceptable values, therefore the domain of the given function is:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \{x\,|\,x\,\in\,\R,\ x\,\neq\,-3\}]


Next time you post a problem, use parentheses to make it clear what things go together.  The way you posted your problem it was equally likely that you meant either:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ y\ =\ \frac{5x}{3x\ +\ 9}]


or


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ y\ =\ \frac{5x}{3x}\ +\ 9]


Had you written:  y = 5x/(3x + 9) your meaning would have been clear.


John
*[tex \LARGE e^{i\pi} + 1 = 0]
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