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- clara1223

Given f(x)=-3x+(1/x)-3, simplify (f(x+h)-f(x))/h when x+4

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- clara1223

Given f(x)=-3x+(1/x)-3, simplify (f(x+h)-f(x))/h when x+4

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- Loser66

Let me help you write it in neat
\(f(x) = -3x+\dfrac{1}{x}-3 \)
Simplify\( \dfrac{f(x+h)-f(x)}{h} \) when x = 4

- Loser66

Where are you stuck?

- clara1223

@Loser66 I don't even know where to start...

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- Loser66

from f(x), make the right hand side has the same denominator, what do you get?

- clara1223

This is what I brainstormed so far, but I am not sure if this is anywhere close to the right track. Plus, it doesn't look like any of the answer choices

- clara1223

- Loser66

Yes, but you make the same denominator wrong!!
redo it here, please

- clara1223

Which denominator are you talking about?

- Loser66

ok, let make it clear
\(f(x) = -3x +\dfrac{1}{x}-3=\dfrac{-3x^2-3x+1}{x}\) ok?

- Loser66

Now, \(f(x+h) = \dfrac{-3(x+h)^2-3(x+h)+1}{x+h}\)

- Loser66

Open parentheses and
Take difference \(f(x+h)-f(x) \) , we have
\(\dfrac{-3(x^2+2xh+h^2)-3x-3h+1}{x+h}-\dfrac{-3x^2-3x+1}{x}\) right?

- Loser66

\(\dfrac{-3x^2-6xh-3h^2-3x-3h+1}{x+h}+\dfrac{3x^2+3x-1}{x}\) ok?

- Loser66

REMEMBER, that is just f(x+h)-f(x) , we didn't \(\div h\) yet!!!

- Loser66

Now, we have 2 fractions with different denominator; We need make them have the same denominator by multiply the first term by x, the second term by (x+h)

- Loser66

It becomes
\(\dfrac{-3x^3-6x^2h-3xh^2-3x^2-3xh+x}{x(x+h)}+\dfrac{(x+h)(3x^2+3x+1)}{x(x+h)}\)

- Loser66

Simplify,

- Loser66

sorry, mistake at the last 1, it is -1, not 1 from numerator of the second term

- clara1223

I understand, keep going please, you've been incredibly helpful.

- Loser66

just that, you simplify, then divided by h more, then replace x =4 to get the answer.

- Loser66

hey, barbecue, is there any shorter way?? Please, help.
My way is tooooooooooooooo long and easy get mistake. hehehe.. barbecue @dan815 Please, please

- clara1223

So far I have:
\[\frac{ -3x ^{3}-6x ^{2}h-3xh ^{2}-3x ^{2}-3xh+x }{ x(x+h) }+\frac{ 3x ^{3}+3x ^{2}h+3x ^{2}+3xh-x-h }{ x(x+h) }\]
after I canceled out:
\[\frac{ 3x ^{2}h-3xh ^{2}-h }{ x(x+h) }\]
Then I took h out of numerator:
\[\frac{ h(3x ^{2}-3xh-1) }{ x ^{2}+xh }\]
How am I doing so far?

- clara1223

@Loser66 what's my next step?

- anonymous

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