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)=6x+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 nine X plus three. We have four answer choices for this problem A B C and, and each of these solutions have three separate solutions all with different quantities. So we can begin this problem by first finding F of X plus H. Well, to find F of X plus H, we just need to substitute in the expression X plus H four X from our original F of X function. So with this F of X plus H, we have instead of nine X plus three, we have nine times the quantity of X plus H plus three. Now, we just need to simplify by distributing in the nine. So we see that F of X plus H is just nine X plus nine H plus three. So now we can move on to finding our next solution, which is F of X plus H minus F of X. Well, to find the solution to this expression. We just need to substitute in our, our expression for F of X plus H and we need to substitute in our expression for F of X. So beginning with F of X plus H, we again have nine X plus nine H plus three and now we subtract F of X. So we have minus the quantity of F of X which is nine X plus three. Now we need to simplify by first distributing in the negative sign. So our expression becomes nine X plus nine H plus three minus nine X minus three. And now we need to just simplify our expression by combining like terms. So we see that both of the nine X terms cancel as well as both, both of the threes cancel. So with this, we see that F of X plus H minus F of X in the simplified form gives us just nine H. So now we have found our second solution and we can move on to finding our third and final solution which is finding the expression for the quantity of F of X plus H minus F of X all divided by H. So he, we're just like before we need to substitute in our expression for F of X plus H minus F of X. And then we need to divide that expression by H. So substituting in our expression, we have nine H and this is now divided by H and simplifying, we see that both of the H terms cancel out. So finally, we see our third solution for F of X plus H minus F of X all divided by H is just simplified to nine. So here we have found our three required solutions. And so we are left here with answer choice D where F of X plus H is equal to nine X plus nine H plus three F of X plus H minus F of X is equal to nine H. And finally, the quantity of F of X plus H minus F of X divided by H is equal to nine. 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)=6x+2

Key Concepts

Function Notation and Evaluation

Difference of Function Values

Difference Quotient

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