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Evaluate the difference quotient for the given function. Simplify your answer.

$ f(x) = 4 + 3x - x^2 $ , $ \dfrac{f(3 + h) - f(3)}{h} $

$\frac{f(3+h)-f(3)}{h}=-3-h$

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okay. And this problem, we're going to evaluate this difference quotient for this function. F of X. This kind of problem really helps us prepare for some problems coming down the road in the future. And so what we want to do is substitute three plus age into the equation for X, and that's going to give us four plus three times the quantity three plus age, minus the quantity three plus h squared. So that takes care of f of three plus h. Now we need to do f of three. So we're going to substitute three in the equation for X. We have minus parentheses, four plus three times three minus three squared. And that's all over h. So now we have a whole bunch of simplifying to do, so we'll be doing some distributing and some multiplying by no meals. So in the next step, we have four plus distribute the three, and we have nine plus three age minus. Now I'm going to multiply three plus age by itself, using the foil method that gives me nine plus six H plus H squared minus. Now we have the last grouping in parentheses, and that is four plus nine minus nine. And all of that is still over. H All right, so let's keep going and simplify this. One easy thing we could do is add the nine and the negative nine those at 20 So those cancel. And now we have four plus 9 13 plus three h and then we'll distribute the minus sign here. So we have minus nine, minus six age minus age squared, and we'll distribute the minus sign. Here we have minus four. That's all over H. OK, notice that we have 13 minus nine, minus four. That would be zero. And then let's combine the other like terms. We have three H minus six age, so that would be negative. Three h and then all by itself, we have minus H squared. So we have negative three age minus age squared over H. Okay, now let's notice that we could factor in h out of the numerator. So if we do that, we have hte times a quantity negative three minus age over h Cancel the age on the top of the bottom and all we have left is negative three minus H