Simplify the difference quotient (ƒ(x)-ƒ(a)) / (x-...

Question

Simplify the difference quotient (ƒ(x)-ƒ(a)) / (x-a) for the following functions.

ƒ(x) = 4 - 4x + x²

Accepted Answer

Hi everyone, let's take a look at this practice problem. This problem says for the function F of X is equal to 11 minus 11 X plus X2, calculate the simplified form of the different quotient, which is the quantity of F of X plus H minus F of X. In quantity, divided by H. We're given 4 possible choices as our answers. For choice A, we have H + 2 X + 11. For choice B, we have H + 2X minus 11. For choice C, we have H + 2X. and for choice D, we have H minus 2X minus 11. So, this question wants us to calculate the simplified form of the difference quotient. So we need to calculate the quantity of F of X plus H minus F of X in quantity, divided by H. So we're going to use our function F of X, which is equal to 11 minus 11 X plus X2d, you calculate this quotient. So, this question is going to be equal to for F of X plus H in our function F of X, I'm gonna replace X with X + H. So I'm gonna have 11, minus 11, multiplied by the quantity of X plus H. In quantity Plus the quantity of X plus H in quantity squared. Then we'll have minus F of X, so that's gonna be minus the quantity of 11, minus 11 X. Plus x 2 in quantity. And that whole expression is divided by H. So now what we need to do is simplify our numerator, and we're gonna do the several operations. So, this is going to be equal to 11. Minus, and here we're going to distribute the 11 inside the um quantity of X plus H, so we'll have minus 11 X. -11 H. Then we'll have plus, and here we want to multiply out the quantity of X plus H in quantity squared, and that will give me X2. Plus 2 XH. Plus H squared. And then finally, we want to distribute our minus sign inside the quantity of 11 minus 11 X plus X squad. So that's going to be -11. Plus 11 X. Minus X2. And all of this is still being divided by H. So again, we still need to simplify our numerator, so this is going to be equal to. And here we're gonna look at it term by term. So, we have a term of 11, and we also have -11, so I want to add those two together, they're going to be equal to 0. I have the next term is -11 X, but I also have plus 11 X later on, so those two terms are going to cancel. The next term I have is -11H and nothing cancels it, so I'll write it down. Then we will have a, for the next term, less X2, but also have minus X2, so when to add those two together, they will add to zero. The next term is going to be plus 2 XH and then plus H squared, and all of this is still divided by H. And I noticed that I can cancel out a factor of each in each of our terms in the numerators. So, this is going to be equal to -11. Plus 2 X plus H. And we're just going to rearrange this to match our answer choices, one of our answer choices. So this is going to be equal to H. Plus 2 X minus 11. So, this is the answer that we were looking for. And if I look at my answer choices, the correct answer choice corresponds to answer choice B. So I hope that this explanation has been useful, and I'll see you in the next video.

Simplify the difference quotient (ƒ(x)-ƒ(a)) / (x-a) for the following functions.
ƒ(x) = 4 - 4x + x²

Key Concepts

Difference Quotient

Function Simplification

Calculus and Derivatives

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