evaluate and simplify the expression

[renegade05]

I am having a tough time with this one. I dont know what to plug in where. PLEASE HELP.

\(\displaystyle \frac{f(a+h)-f(a)}{h} \ \ \ if \ \ f(x) = \frac{x}{3x+2}\)

 

[wjm11]

Don't make it too complicated. This is just standard function notation:

f(x) = x/(3x + 2)
f(a + h) = (a + h)/(3(a + h) + 2)
f(a) = a/(3a + 2)

Make these substitutions, then proceed.

 

[renegade05]

i tried to simplify but i cant seem to do it. please help

 

[mmm4444bot]

Will you please show us what you tried ?

I mean, it's hard for tutors to help you when they can't see where you're stuck.

If you're trying to say that you need lessons on simplifying rational expressions, then HERE are some lessons and examples.

Otherwise, let's see your work.

Cheers

 

[renegade05]

Allright this is what i did and where i am stuck.

\(\displaystyle \frac{f(a+h)-f(a)}{h} \ \ \ if \ \ f(x) = \frac{x}{3x+2}\)

\(\displaystyle \frac{\frac{(a+h)}{3(a+h)+2} - \frac{a}{3a+2}}{h}\)

\(\displaystyle \frac{\frac{(3a+2)(a+h) - a(3a+3h+2)}{(3a+3h+2)(3a+2)}} {h}\)

\(\displaystyle \frac{\frac{3a^2+3ah+2a+2h-3a^2+3ah+2a}{12a+9a^2+9ah+6h+4}} {h}\)

\(\displaystyle \frac{\frac{6ah+4a+2h}{12a+9a^2+9ah+6h+4}} {h}\)

\(\displaystyle \frac{6ah+4a+2h}{12ah+9ha^2+9ah^2+6h^2+4h}\)

And thats where i got to. I might be doing this completely wrong or some trick i dont know about. I have been out of high school for 5 years and just decided to get back into college math...so... you know. "Use it or lose it."

Please help.

 

[renegade05]

anyone?

 

[mmm4444bot]

\(\displaystyle \frac{\frac{3a^2+3ah+2a+2h-3a^2+3ah+2a}{12a+9a^2+9ah+6h+4}} {h}\)

You missed some signs on the very top.

-a(3h) = -3ah

-a(2) = -2a

Therefore, the upper numerator simplifies to 2h.

Also, it's generally a good idea to leave denominators in factored form (to be able to see potential future cancellations).

Lastly, we can avoid the compound ratio by first factoring out the division by h.

Dividing by h is the same as multiplying by 1/h.

\(\displaystyle \frac{1}{h} \cdot \frac{2h}{(3a + 3h + 2)(3a + 2)}\)

Can you do the final step ?

 

[renegade05]

stupid signs!

my final answer i get:

\(\displaystyle \frac{2}{12a+9a^2+9ah+6h+4}\)

correct?

 

[mmm4444bot]

Yes, your result looks okay.

My simplification would look like this:

\(\displaystyle \frac{2}{(3a + 3h + 2)(3a + 2)}\)