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$g(x) =\frac{3-5x}{5-x}$

I know that I am retarded for not being able to do this on my own. Algebraically you just have to solve for x. Even so, I can't do it.

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    You're not retarded if can't solve something. Generally in mathematics you need a lot of practice when learning something so that there will come a time when you can see that problem as "easy", but the thing is that one usually tends to forget how many hours were spent practicing and solving exercises to get to that stage. Obviously there are things that cannot be considered easy no matter how much you practice or study.2010-11-20
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    Suggestion: Begin by multiply both sides by $5-x$ (which you can do provided that $5-x\neq 0$). Then transform your equation into the form $(cg(x)+d)x=(ag(x)+b)$ by adding or subtracting suitable terms to both sides. Finally solve for $x=\dfrac{ag(x)+b}{cg(x)+d}$. This expression is valid for $cg(x)+d\neq 0$. The inverse function is $g(x)\mapsto \dfrac{ag(x)+b}{cg(x)+d}$2010-11-20
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    I edited your title; no need to beat down yourself in front of us.2010-11-20

2 Answers 2

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Hint: $y=\frac{3-5x}{5-x}$ so $y(5-x)=3-5x$. Now collect terms involving $x$ on one side and those not involving $x$ on the other and solve for $x$.

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    That part I get, but from there an out I mess it up.2010-11-20
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    We can write $5y-xy=3-5x$. So, $5y-3=x(y-5)$. Can you solve for $x$ now?2010-11-20
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Maybe you could start with a less complicated example and build up from there. Why don't you try finding the inverse function of $f(x) = x + 3$ using the idea that Timothy gave you of putting $y = x + 3$ and then solving for $x$. Then you can try $f(x) = 7x - 2$ for instance. Now you could also try $f(x) = \frac{3}{x}$ for example.

Something that is sometimes very useful is to consider "easier" problems related to your original problem and trying to solve those.

I believe the point is to gain some experience on what the particular difficulty may be, and that the "harder" problem may combine some of the difficulties that you will have treated in the particular, less complicated examples that you considered. This advice is treated by Polya in his excellent book about problem solving.

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    I can to do those examples, that's why this problem makes me so angry.2010-11-20