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Given $f(x, y)=x^{2}-2 x y^{2}-3 y^{3},$ determine (a) $f(2,-1),(\mathrm{b}) f(-1,2),$ if neither $h$ nor $k$ is zero, determine (c) $\frac{f(x+h, y)-f(x, y)}{h}$, (d) $\frac{f(x, y+k)-f(x, y)}{k},(\text { e }) \frac{f(2+h, y)-f(2, y)}{h}$,(f) $\frac{f(x, 1+k)-f(x, 1)}{k}$. (g) In (c) and (e) what happens if you allow $h$ to equal 0 at the end of your calculations? (h) Same question for (d) and (f) if you allow $k$ to equal 0 at the end of your calculations.

(a) 3(b) -15(c) $2 x-2 y^{2}+h$(d) $-4 x y-2 x k-9 y^{2}-9 y k-3 k^{2}$(e) $4+h-2 y^{2}$(f) $-4 x-2 x k-9-9 k-3 k^{2}$(g) $2 x-2 y^{2}, 4-2 y^{2}$(h) $-4 x y-9 y^{2},-4 x-9$

Algebra

Chapter 1

Functions and their Applications

Section 2

Basic Notions of Functions

Functions

Oregon State University

University of Michigan - Ann Arbor

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).

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for this problem. We're going to do several things with this function. Here we have a function in two variables X squared minus two X y squared minus three y cute. So first were the first to problems We're going to just plug numbers in. So I'm going to take my function and first I'm going to do it with X equaling two. And why equals negative one? Well, anytime we have a function where we have something that we've changed here we have just an X and a y in our original function. But here we're doing a substitution because it's two variables we're doing. A double substitution X is gonna become, too. Why is going to become negative one? So let's plug those values into our original function. Access to that could be two squared times too. Well, excess to why is negative. Once I plugged both of those in and I have minus three times negative. Why, Cube? Just the only thing you really have to be careful with these, uh, to variable functions is make sure you're plugging the right value in for the right variable. In this case, two for X, negative one For why Okay, So if we simplify this, that gives me four. My second piece is going to be minus four and then plus three, which gives me a value of three on the other end. Okay, what if I switch them? What if I say f of negative 12? Well, same numbers, But again, you have to be careful. What? You're putting them in the right place now. Negative one goes in for X two. Goes in for Why? So that's going to give me negative. One square. There's my X squared minus two times X negative. One times. Why Squared two squared. And that's gonna be minus three. Too cute. So one. The second piece here. I've got two negatives that's going to be a positive two times four. That's going to be eight and then minus three times eight. That's 24. When I do all of that out, I'm going to get a negative 15. Okay? And if I picked any two numbers, we would substitute those in exactly the same way. Now what happens, though, if I put something in this not just a plain number, for example, what if I have f of X plus h and why? Well, now, when I am sorry and I'm actually gonna make a little bit bigger than this, I would take f of X plus h y minus f of X y divided by h. So I've got two calls. The first the second one here F of X y is just the function that I have F of X plus a h means X plus h is going to replace the plain X in my original function. So what does that look like? Well, I have X squared that's now going to be X plus H squared minus two x y squared will now axes X plus h and then minus three y cube notice. I just have a Y that I'm substituting in. So the white pieces actually aren't changing. And then I'm gonna subtract f of X y. So, um, subtracting X squared minus two X Y squared minus three Y cube over H. Well, let's see what we can simplify here. I'm gonna expand everything out. Try to get rid these parentheses first, I have X squared plus two x h plus H squared next minus two X y squared minus two h Y squared minus three y Cube. And now we distribute that negative. We get negative. X squared plus two x y squared plus three y cube over each. Okay, now we have some things to cancel. I have a x squared minus X squared negative two x y squared plus two x y squared minus three y cubed plus three y cube. Also, every term that's left has an h so I could get rid of h here. I get rid of one of those. H is I get rid of the h here and that cancels with the H and the denominator. So what I have is two x plus h minus two y squared. That's coming from this term this term in this term. Okay, so it's a little bit longer, but it's because I'm keeping the variables in the in their places here. It's a little bit simpler when it's just numbers. Okay, Next, let's try. It's kind of the same thing, Onley. This time I'm gonna keep the X by itself and I'm going to change the why toe y plus k and again it's gonna be the same similar thing f of x y and this time I'm dividing by K. Since this case what? I'm putting in the numerator. Okay, Just like we did before. I'm gonna plug them into that formula. X squared minus two x y squared minus three Y cute. So when I do that, I'm gonna have X squared minus two x. Why squared now? My, Why is why plus K minus three? Why cubed and again? Why is why? Plus K and I want to subtract X squared minus two X y squared minus three y cute. This is all going over, K scroll up. So I've got a little more room and let's get rid of all our parentheses here. I have to expand that y plus k. So that's going to be actually, let's let's do that out over here. Why? Plus, K when I expand, it is why squared plus two y k plus K squared. So I will have two x y squared minus four x y que Remember, we're multiplying this by negative two x minus two x k squared. Now let's remember how we expand out something Cube. This will be white, cubed plus three y squared K plus three y k squared, plus k cubed and I'm multiplying it all by negative three. So negative three y cubed minus nine y squared K minus nine y k squared minus three K cube minus every term here so minus X squared plus two x y squared plus three y cubed Who? That is a lot of algebra But it all works out because now we've got some things that can cancel I haven't x squared and a minus x squared I have a negative two x y squared plus two x y squared negative three y cubed plus three y cute. Okay, that leaves me. I'm just going to circle these so they stand out. I've got 123 for five terms that did have a square that I kind of wrote over there, each one of which has a case I'm gonna be able to divide it by that K, uh, in my denominator. So when I do that, I get negative for X Y minus two xk, remember, I'm dividing out. Okay. Here minus nine y squared minus nine y que minus three k squared. So that's my final answer for that piece. Hey, next we'll hear about two more of these were going to do, and then we've got something we're going to just kind of compare at the very end. So let's pick out a new color this time around. We're going to a combination of numbers and letters were going to f of two plus h Why, oops! Helps fight right it properly f of two plus h y minus f of two y over h. So again, just as a reminder. I'm gonna just kind of write this over here so we can see it. Our function is X squared minus two x Y squared minus three. White Cube. Don't forget, that's our function is way up at the top here. Hey, So when I go to plug this in, we have X squared in the place of X. We now have two plus h so two plus h squared minus two times two plus h y squared minus three y cubed. And I'm going to subtract X squared minus two X y squared minus three y cubed all over h. So let's get rid of some of those parentheses. First, we square four plus four h plus H squared minus four y squared minus two h y squared minus three y Cube. And now we get rid of that. Those parentheses, minus X squared plus two x y squared plus three y cubed over each. Okay, well, what cancels? I have a see, I have a negative three y cubed in a positive three y cube, but and I need to apologize. I made a mistake here on the line before, so I'm gonna go back and fix this. Um, I just put X and y, and I shouldn't have because it's too. And why so X squared is really supposed to be two squared or four. So I fixed that one and my two x here should also be a four. My apologies was talking was not paying attention. So down on this line, it should be minus four plus four y squared. Now, this is how things were going to cancel. So I have a positive for in a negative four. I have a positive four y squared in a negative four y squared. That leaves me with one, 23 terms, each of which has an h so I can cancel that age with my denominator and get four plus h minus two y squared. Okay, to one more of these f of Now, we're gonna leave the X, but it's going to be one plus K minus f of X and one all over K. Okay, so this could look really familiar. Now we're just plugging in these values, so X doesn't change. I've got X squared minus two X. Well, my why is one plus k squared minus three times one plus k cute. And I'm going to subtract f of X one. So that's X squared. Why is one so that's minus two X. Why is one so that's minus three all over. Okay, so let's expand this out. X squared. Well, I'm multiplying one plus k squared by negative two X and that's going to give me you see what I have here. Okay, so it could be minus two X minus four X K minus two x k squared. Now, next piece, I'm gonna be expanding that cube and multiplying everything by negative three. So that gives me negative three minus nine K squared, minus nine K minus three K cube. And now we're going to get rid of this negative here in these parentheses. So it's minus X squared plus two x plus three all over. Okay, so let's see what cancels. I haven't X squared minus X squared. Negative two X positive two X negative three plus three. That leaves me one to 345 terms, all of which have a case I'm going to cancel with that K in the bottom. That gives me minus four X minus two X K minus nine K minus nine, minus three K squared. Now we're going to go back and look at the last four. And the question is, in each of these last four cases, if we let either h or K, depending on which variable we have if we let them equal zero what's the value of our function then? So we're gonna go up here. We're gonna start with our green one. If I let H equals zero, then what I have here is two x minus two y squared from my blue function. If I let h or this case K if I like K equals zero, I end up with negative for X Y minus nine. Why squared the other terms? I'll go to zero if K is zero. How about my red function. Well, if H equals zero, then I lose that middle term. It's just four minus two. I squared and from my last one here, if I let K equals zero, that means I'll have negative for X minus nine. Every other term will be zero if K is zero. So here's my function and a variety of ways that we have found different values for f of X and Y.

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