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Given $f(x, y, z)=x^{2}-2 x y^{2}-3 y^{3} z^{2}+z^{2},$ determine (a) $f(1,-2,3)$(b) $f(0,1,-2)$ if neither $h, k, l$ is zero, determine (c) $\frac{f(x+h, y, z)-f(x, y, z)}{h}$. (d) $\frac{f(x, y+k, z)-f(x, y, z)}{k}$, (e) $\frac{f(x, y, z+l)-f(x, y, z)}{l}$.

(a) 218(b) -8(c) $2 x+h-2 y^{2}$(d) $-4 x y-2 k x-9 y^{2} z^{2}-9 k y z^{2}-3 z^{2} k^{2}$(e) $-6 y^{3} z-3 y^{3} l+2 z+l$

Algebra

Chapter 1

Functions and their Applications

Section 2

Basic Notions of Functions

Functions

Missouri State University

Harvey Mudd College

Baylor University

Idaho State University

Lectures

01:43

In mathematics, a function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output. An example is the function that relates each real number x to its square x^2. The output of a function f corresponding to an input x is denoted by f(x).

03:18

01:51

Let $f(x, y, z)=x^{2} y^{4…

01:49

01:29

$f(x, y, z)=2 x^{2}-3 x y^…

11:04

Let $f(x, y, z)=x^{2} y+x …

04:16

$$\begin{aligned}&…

01:45

02:35

Let $F(x, y, z)=x^{3} y z^…

03:49

Let $f(x, y, z)=x^{3} y^{5…

05:52

for this problem. We've been given a function in three variables X, y and Z. Now, one thing it's important to remember when we're going to go plug in values. For example, I'm gonna take our first case here. F of one negative 23 We have three values here. The first one is X. The second is why the third Izzy. The most common place to make mistakes when you're doing these is that you plug in the wrong number for the wrong variable. So just be a little extra careful. Make sure that in this case, you know, one is always in for X negative two for why three frizzy? Um, don't get them mixed up. Just be careful. Um, you pay attention to the details, so let's do a couple here with just numbers. If I have f of one negative 23 that means I'll have one squared minus two times one times ticket of two squared minus three times negative. Two cubed times three squared plus three squared. So we just plugged in all these numbers all the way across. And if I plug those into a calculator, I confined that. My answer is 218. Let's try one more of these with just numbers. How about f of 01 negative, too? Well, if x zero that's zero squared minus two times zero times negative. Two squared. The good thing you could see already. Both of these terms are gone. They're zeros. So we really have left for the last two terms. Three times, one cubed times negative two squared plus negative two squared. And that gives me a final value of negative eight. Okay, Now let's try a couple of these where we're gonna have letters and not numbers that we're plugging in. We're going to do once for each of the three letters. So first, instead of X, we're gonna plug in X plus H Y and Z, and I'm gonna subtract from that f of x, y and Z over h. The algebra does get a bit long on these again. Be careful. Pay attention to the details, just slog through. It will. A lot of things will cancel, and the dancers aren't that. You know, the answers are better than it looks in the middle of the process. So let's get started. I have X squared. Well, X is now X plus H minus two times X y squared again X X plus H minus three y cube Z squared plus Z squared. And I'm going to subtract F of Z, which is what I have above, and the whole thing will be divided by H. Now I'm gonna want to get rid of some of these parentheses, expand things out. But first, there's a few things I can cancel right off the bat. These last two terms did not have an X in them, so they are identical. So I'm just attracting themselves. They're canceled immediately. I don't have to copy them any further. Now let's get rid of our parentheses. First, I've got X squared plus two x h plus H squared. Now I'm going to distribute two y squared. So I've got minus two x y squared minus two h y squared, and I'm going to subtract X squared. It's plus two x y squared over each. Well again, I've got some things to cancel. X squared. Those canceled negative two x y squared, and it's positive they cancel. Every other term has an H in it, which I can cancel with the denominator so that gives me two X plus H minus two Y squared. Okay. Next. Come down just a little bit. We do the same thing. But now I'm gonna leave the X by. It's by alone, and I'm gonna have why plus k in place of why. And again, I'm gonna set it up the same way minus f of X y z. And this time since I'm adding K, I'm gonna divide by K. So let's plug these in. I have X squared minus two x y squared. Remember? Why is now why plus k minus three y cubed z squared again? Why is why plus K plus c squared and I'm going to subtract my functions squared minus two x y squared minus three y cubed Z squared plus z squared all over. Okay, again, I know the algebra is long, but hang in there with me. Okay. What can cancel? Well, I know that the first and last terms did not have a why They're exactly the same. I can cancel those now. We want to get rid of our parentheses. We have negative two X. I'm gonna do these in two steps here because I've got a square and a cube. So let me expand these out. Next step will get rid of the parentheses altogether. This is why squared plus two y que plus k squared. And then I have minus three z squared. Don't forget that Z squared on the back end there. This is why cute plus three y squared K plus three y k squared plus k cute. And I'm subtracting thes last terms, which means I'm just changing the signs. That's plus three. Wyke. Yeah, plus and plus on both of them. Okay. All over. Okay. Now, let's get rid of our parentheses. Negative two x y squared minus four X Y que minus two x k squared. All of these are gonna be negative. Minus three z squared y cubed minus nine z squared. Why squared K minus nine z squared. Why? K squared minus three. Z squared. Que cute. Got these last two terms hanging on the end here. Okay. What cancels? Well, I have a positive and negative for those. And ah, positive and a negative here. Every other term has a K in it, so I can cancel with that k and the denominator. And that leaves two B with negative four x y minus two x k minus nine Z squared. Why? Squared minus nine z squared y que this is? Yep, that's correct. And then minus three Z squared K squared. Okay, that was quite a long one. One more to go for this last one. I'm gonna leave X and y alone. And now I have Z plus l again. Same procedure minus F of X, Y and Z. And this time, since I'm adding l on the top, I'm gonna be dividing by l. So let's plug things in. X squared minus two x y squared minus three y cubed Z squared. And remember Z is Z plus l plus c squared again. Z plus l I'm going to subtract X squared minus two x y squared minus three y cube z squared plus z squared all over elf Well, like before. Let's cancel what we can. The first two terms did not have a Z. They're identical. So they're gone When I expand out my parentheses Here I get negative. Three y cube Z squared. I'll have a to z else that becomes negative. Six y cubed cl minus three y cubed l squared that has come expand our next one z squared plus two l z plus l squared. And when I subtract these last two things, they're going to be positive. Well, that's a positive. And this one's a negative all over l Well, let's see what cancels. I have a negative and a positive. They cancel positive and a negative. They cancel everything that's left has an L that canceled with my denominator. So I end up with negative six y Cube Z minus three. Why cubed l plus two z plus l. And that's my final answer.

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