Complex Fractions: simplify [f(x+h)-f(x)]/h for f(x)=(1-x)/x

[MaxRabbit]

No idea how to write this perfectly, but here goes-
Directions: Express [f(x + h) - f(x)] over [h] as a single simplified fraction.
A sample problem "1)": f(x) = [1 - x] over [x]
I don't really know how to set it up; I tried like: [{(1 - x) over (x + h)} + {(1 - x) over (x)}] over [h] but that didn't seem to work...

Another similar problem that I don't understand, with directions just to simplify, is this one:

A sample problem "2)": [u + (1 over 1 + {1 over u})] over [(1) over (u + 1)]

Thanks for help with either problem.

 

[pka]

Re: Complex Fractions

Here is a start for you.
\(\displaystyle f(x) = \frac{x}{{x + 1}}\)

\(\displaystyle \frac{{f(x + h) - f(x)}}{h} = \frac{{\frac{{x + h}}{{x + h + 1}} - \frac{x}{{x + 1}}}}{h} = \frac{{\left( {x + 1} \right)\left( {x + h} \right) - x\left( {x + h + 1} \right)}}{{h\left( {x + h + 1} \right)\left( {x + 1} \right)}} \\ \end{array}\)

 

[MaxRabbit]

Re: Complex Fractions

Okay, but 1) was like this: \(\displaystyle f(x) = \frac{1 - x}{{x}}\)

 

[jwpaine]

Re: Complex Fractions

Okay, but 1) was like this: \(\displaystyle f(x) = \frac{1 - x}{{x}}\)

So what you are saying is that you expect us to do the work out for you? Pka evaluated a very similar function by the difference quotient. That should be enough to get you started.

Surely you know function composition?

John

 

[MaxRabbit]

Re: Complex Fractions

Okay, but 1) was like this: \(\displaystyle f(x) = \frac{1 - x}{{x}}\)

Pka evaluated a very similar function by the difference quotient. That should be enough to get you started.

Surely you know function composition?

I do, and I pretty much get the ones with two numbers in the denominator-but it's especially the ones with the two terms in the numerator that confuses me