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

Question

For each function, find (a)ƒ(x+h), (b)ƒ(x+h)-ƒ(x), and (c)[ƒ(x+h)-ƒ(x)]/h. ƒ(x)=1/x

Accepted Answer

Hey, everyone Hubert 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 divided by X. Now we have four different answer choice options for this problem, A B C and D and for each answer choice, we have three different solutions. One for each of the expressions. So we can begin by finding the solution for our first expression F of X plus H. While to find F of X plus H, all we need to do is substitute in X plus H for X from our original function. So instead of having two divided by X, we now have two divided by the quantity of X plus H. Now there isn't any other simplifying that we can do with this expression. So this is our final answer for F of X plus H. Now we can move on to finding our next expression which is F of X plus H minus F of X. Well, to find this expression, all we need to do is substitute the expression we already found for F of X plus H as well as substitute in the given expression for F of X. So starting with F of X plus H again, we had two divided by the quantity of X plus H and now we have minus F of X. So we have minus two divided by X. Now, we need to simplify this expression by combining our fractions. But before we can combine the fractions, we need to get common denominators for both of the fractions. So we need to multiply our first fraction by the denominator of the second fraction. So two divided by the quantity of X plus H gets multiplied by X divided by X. And now we repeat this for our second fraction but instead multiply the second fraction by the first fractions denominator. So two divided by X gets multiplied by X plus H divided by X P plus H. Now we just need to simplify this expression. So we have two X divided by X times the quantity of X plus H minus two times the quantity of X plus H all divided by X times the quantity of X plus H. Now we can simplify this expression further by combining the enumerators since both of our fractions have common denominators. So combining the enumerators and distributing in the negative two, we have two X minus two X minus two H all divided by X times the quantity of X plus H. So now with this, we just need to simplify this expression further by combining the like terms in the numerator and in the numerator, we see that both of the two X terms cancel out. So our final simplified expression for F of X plus H minus F of X is negative two H all divided by X times the quantity of X plus H. So now that we have found the second expression or the second solution to our second expression, we can move on to finding the third expression which is the quantity of F of X plus H minus F of X all divided by H. So now we just repeat the same process as we did previously. So we substitute in the given or known expression. So we already found F of X plus H minus F of X. So all we need to do is substitute in that expression and divided by H. So we have negative two H divided by X times the quantity of X plus H. And this is all divided by H. Well, we know that dividing by a value is the same as multiplying by the reciprocal. So rewriting this expression, we have negative two H divided by X times the quantity of X plus H time times one divided by H. And now we see we have an H in our numerator and this H in the denominator of this second fraction. So both of the HS cancel out. And now our final simplified form for the quantity of F of X plus H minus F of X divided by H is the simplified form of negative two divided by X times the quantity of X plus H. So with this, we have found our three separate solutions for each of our expressions here. So with these solutions, we are left with answer choice. A where F of X plus H is two divided by X plus H and F of X plus H minus F of X is negative two H divided by X times the quantity of X plus H. And finally, the quantity of F of X plus H minus F of X all divided by H is negative two divided by X times the quantity of X plus 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. ƒ(x)=1/x

Key Concepts

Function Notation and Evaluation

Difference of Function Values

Difference Quotient

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