For each function, find (a)ƒ(x+h), (b)ƒ(x+h)-ƒ(x), and (c)[ƒ(x+h)...

Question

For each function, find (a)ƒ(x+h), (b)ƒ(x+h)-ƒ(x), and (c)[ƒ(x+h)-ƒ(x)]/h.See Example 4. ƒ(x)=-x^2

Accepted Answer

Hey, everyone here we are asked to solve for F of X plus H F of X plus H minus F of X and the quantity of F of X plus H minus F of X divided by H for the following function F of X is equal to negative four X squared. So for each of our expressions that we need to find, we have four different answer choice options, A B C and D and for each answer choice, we have three different solutions. One for each of our expressions. Now we can start with finding our first expression of F of X plus H. Well, to find F of X plus H, all we need to do is substitute in the expression X plus H four X from our original given function. So instead of having negative four X squared, we will now have negative four times the quantity of X plus H squared. So now we just want to expand this expression by first factoring out the quantity of X plus H squared. So doing this, we have negative four times the quantity of X squared plus two X H plus H squared. And now we just want to distribute in this negative four. And we see our final expression for F of X plus H becomes negative four, X squared minus eight X H minus eight X H minus four H squared. And there aren't any other simplifications that we can make here. So this is our final expression for F of X plus H. And so now we can move on to our second expression where we have F of X plus H minus F of X. So to find this expression only we need to do is substitute in the expression we just found for F of X plus H as well as a substitute in the given expression for F of X. So beginning with F of X plus H again, we have negative four X squared minus eight X H minus for H squared and now we have minus F of X. So we have minus the quantity of negative four X squared. So now we just need to combine our like terms to simplify this expression. So first, we just need to distribute in this negative sign. And so we have negative four X squared minus eight X H minus for H squared plus four X squared. And now combining like terms, we see that both of the four X squared terms cancel out. So our final expression for F of X plus H minus F of X simplifies to negative eight X H minus four H squared. And now our last step here is actually to factor out H. So when we factor out H our expression for F of X plus H minus F of X is H times the quantity of negative eight X minus four H. So now that we have found our simplified expression for our second expression, we can move on to finding the quantity of F of X plus H minus F of X divided by H. So now we can repeat a similar process as we did previously by substituting in the known information. So we just found this numerator which is F of X plus H minus F of X. So we can just substitute in that expression that we found and divided by H. So we have H times the quantity of negative eight X minus four H all divided by H. And now we see we have an H in the numerator and the denominator. So both of the H terms cancel. So with this, our final expression for the quantity of F of X plus H minus F of X divided by H becomes just negative eight X minus for H. So with this, we have found our three separate solutions, one for each of the expressions. And with these solutions, we are left here with answer choice D where F of X plus H is negative four X squared minus eight H X minus four H squared. And then F of X plus H minus F of X is H times the quantity of negative eight X minus four H. And finally, the quantity of F of X plus H minus F of X divided by H is negative eight X minus four H. Thanks for watching. I hope you found this video helpful and I'll see you again next time.

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For each function, find (a)ƒ(x+h), (b)ƒ(x+h)-ƒ(x), and (c)[ƒ(x+h)-ƒ(x)]/h.See Example 4. ƒ(x)=-x^2