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 two X squared. Now, for this problem, we have four answer choice options, A B C and D and each of these answer choices have three different solutions. One for each of the expressions that we need to find. So we can begin by finding our first expression F of X plus H. Well, to find F of X plus H, all we have to do is substitute in this expression X plus H for X from our original given function. So instead of having two X squared, we now have two times the quantity of X plus H squared. So we want to expand this expression by first factoring out this quantity of X plus H squared. And doing this, we have two times the quantity of X squared plus two X H plus H squared. And now we just need to distribute in the two. And we see that our expression for F of X plus H is just two X squared plus four X H plus two H squared. And now that we have found our first expression, there isn't any other simplifications we need to do. So this is our final expression for F of X plus H. And now we can move on to our second expression which is F of X plus H minus F of X. So now here to find this expression, we just need to substitute in the expression we just found for F of X plus H and then we need to see substitute in the given expression for F of X. So starting with F of X plus H again, we have two X squared plus four X H plus two H squared and then we have minus F of X. So we have minus two X squared. Now we need to simplify this expression by first combining the like terms. And we see that both of the two X squares cancel out. So we have left, we are left with four X H plus two H squared. And now our next simplification we want to do here is just factor out and H from both terms. So with this, we see our final expression for F of X plus H minus F of X. After factoring out the H term becomes H times the quantity of four X plus two H. So now we have found our second expression that we needed to find and simplified it appropriately. So we can move on to our third expression where we have the quantity of F of X plus H minus F of X divided by H. So now we just need to repeat a similar process as previously by substituting in our known information. So here we just found this F of X plus H minus F of X. So we can just substitute in this expression and divided by H. So doing this, we have H times the quantity of four X plus two H all divided by H. Now we see that we have an H in the numerator and in the denominator. So both of these terms cancel. Now we can rewrite our simplified expression for the quantity of F of X plus H minus F of X divided by H. And we see that this simplified expression is just four X plus two H. So with this, we have found our three different expressions or three different solutions for each of our given expressions. And so we are left here with answer choice B where F of X plus H is two X squared plus four H X plus two H squared. And then F of X plus H minus F of X gives us eight times the quantity of four X plus two H. And finally, the quantity of four X plus H minus four X divided by H is equal to four X plus two H Thanks for watching. I hope you found this video helpful and I'll see you again next time.

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

Key Concepts

Function Notation

Difference Quotient

Limit Concept

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