Simplify the difference quotients ƒ(x+h) - ƒ(x) / h&nb...

Question

Simplify the difference quotients ƒ(x+h) - ƒ(x) / h and ƒ(x) - ƒ(a) / (x-a) by rationalizing the numerator.

ƒ(x) = √(1-2x)

Accepted Answer

Hello there. Today we're going to solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. For the function F of X is equal to the square root of 5 minus 7 X. calculate the difference quotients. F of X plus H minus F of X all divided by H. And F of X minus F of A all divided by X minus A. Awesome. So it appears for this particular problem, we're asked to calculate the difference quotients for F of X plus H minus F of X all divided by H. And F of X minus F of A all divided by X minus A. So we're ultimately trying to solve for two separate answers using our basic function F of X is equal to the square root of 5 minus 7 X. So ultimately, what we're trying to do is we're trying to calculate these difference quotients for these two separate expressions. So we're trying to solve for two separate answers. So with that said, Let us start to solve for this problem by first focusing our attentions on F of X plus H first. So we're solving for F of X plus H first. So let us recall and note that F of X plus H will be equal to the square root of 5 minus 7 multiplied by X plus H, because that's what we're plugging in for our value of X. Which will be equal to when we simplify the square root of 5 minus 7 X minus 7H. Fantastic. So now we need to substitute in the required values into our formula, which, as we should recall, our formula is F of X plus H minus F of X, all divided by H. So we need to plug in. Our, so essentially we're plugging in, we're substituting in our required values into our formula. So when we do that, we will find that F of X plus H minus F of X, all divided by H is going to be equal to the square root of 5 minus 7 X minus 7H. And we need to take this, and we need to subtract the square root of 5 minus 7 X all divided by H. And this will all be equal to the square root of 5 minus 7 X minus 7H minus the square root of 5 minus 7 X. All divided by H. Multiplied by the square root of 5 minus 7 X minus 7 H plus the square root of 5 minus 7 X. All divided by the square root of 5 minus 7 X minus 7H. Plus the square root of 5 minus 7 X. Awesome. So, this will all equal. Drum roll, 5 minus 7 X minus 7 H minus 5 + 7 X. All divided by H multiplied by the square root of 5 minus 7 X minus 7H. Plus, the square root of 5 minus 7 X. Which will be equal to Negative. 7 H divided by H multiplied by the square root of 5 minus 7 X minus 7H. Plus, The square root of 5 minus 7 X. Which will ultimately equal when we simplify one last time, negative 7 divided by or 7 all divided by 5 minus 7. Multiplied by X plus H. Plus the square root of 5 minus 7 X, and this is equal to F of X plus H minus F of X all divided by H. and that is our first final answer. Hooray, we did it. So we found one of our two answers. Now we need to solve for our second answer. So now we need to focus our attention now on F of A. So now we need to find F of A. So switching gears now to focus our attention on F of A, we need to recall and note that F of A is equal to the square root of 5 minus 7, plugging in a value of A N for X. Awesome. So now we need to substitute in our required values into our formula, which, as we should recall, our formula states that F of X minus F of A. All divided by X minus A. Awesome. So, using our formula, we need to say that F of X minus F of A, all divided by X minus A, is equal to the square root of 5 minus 7 X. And we need to take this, and we need to subtract the square root of 5 minus 7A. All divided by X minus A. Fantastic. So now our next step is we need to say that this is equal to the square root of 5 minus 7 X minus the square root of 5 minus 7A, all divided by X minus A, which is equal to or actually not equal to. So we need to take this expression, which is the square root of 5 minus 7 X minus the square root of 5 minus 7A, all divided by X minus A. And we need to multiply it by the square root of 5 minus 7 X plus the square root of 5 minus 7A. All divided by the square root of 5 minus 7 plus. The square root of 5 minus 7A. And this will be all equal to Drum roll, so this will all be equal to. 5 minus 7 X minus 55 plus 78. So, let's repeat that again since I went a little fast here. So this will be all equal to 5 minus 7 X minus 5 + 7A, all divided by X minus A. And then we need to take X minus A, and we need to multiply it by the square root of 5 minus 7 X plus. The square root of 5 minus 7A, fantastic. And this will be all equal to negative. 7 multiplied by 7, sorry, sorry, so it's gonna be all equal to negative, 7 multiplied by X minus A, all divided by X minus A multiplied by the square root of 5 minus 7 X plus the square root of 5 minus 7A. Fantastic. And now, We need to say that this will ultimately be all equal to negative. 7 divided by the square root of 5 minus 7 X plus the square root of 5 minus 7A. Awesome, and that's it. That is our final answer. Hooray, we did it. So ultimately, essentially, our final answer for our second answer. So for F of X minus F of A, all divided by X minus A, it will be equal to -7 divided by, so 7 all divided by the square root of 5 minus 7 X plus the square root of 5 minus 7A. And that is it. That is our final answer. We've officially solved for this problem. Hooray, we did it. So quickly recapping here what we've learned. So ultimately, our final answers From the top, that for F of X plus H minus F of X, all divided by H will be equal to -7, all divided by the square root of 5 minus 7. Multiplied by X plus H plus the square root of 5 minus 7 X. And for F of X minus F of A all divided by X minus 8, it will be equal to -7, all divided by the square root of 5 minus 7 X plus the square root of 5 minus 7A. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

Simplify the difference quotients ƒ(x+h) - ƒ(x) / h and ƒ(x) - ƒ(a) / (x-a) by rationalizing the numerator.
ƒ(x) = √(1-2x)

Key Concepts

Difference Quotient

Rationalizing the Numerator

Function Composition

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