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Let $y=f(x)$ describe the upper half of the circle $x^{2}+y^{2}+8 x-6 y+9=0 .$ Determine (a) $f(-3),$ (b) $f(-6),$ (c) $f(-1)$

(a) $3+\sqrt{15}$(b) $3+2 \sqrt{3}$$(c) 3+\sqrt{7}$

Algebra

Chapter 1

Functions and their Applications

Section 5

The Circle

Functions

McMaster 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 have been given the equation of a circle and we want to be looking at a function. But we have a problem. A circle is not a function. And just as a reminder, this is just a generic circle. I'm not even gonna put it on a grid. But if you have a circle, this does not pass the vertical line test. If I put a vertical line through, I'm gonna hit my circle in more than one place. So circles aren't functions. So what we have done in order to be able to create a function here, is we're going to look a Onley part of the circle in particular. We're going to look at the upper half of the circle, so that means we erase my circle raised by circle, but a fresh one up here. So if I have a circle, the upper half of the circle means I'm only gonna be going from here. Thio here. That red piece is a function. Now. Any vertical line going through that red is on Lee going to hit it in one place. So as long as I only do a half a circle this is going to be a function. So let's take our equation We're going to solve for why? Which will tell us what our function is. Remember, our function equals why? So in order to solve this, let's put this into standard form. We're gonna complete the square, so I'm gonna put my excess together. I'm gonna put my wise together and that positive nine. I'm going to subtract. Move it over to the other side. Now let's complete Are square Half of the extra coefficient is four. We add that 16 and we have to do it to both sides. That keeps us balanced. So that gives me X Plus four squared. Now for my why we look at our why term Take half the coefficient, which is negative. Three square it, which is nine. We have to add that to both sides. So that gives me why minus three squared. And when I add all those numbers on the right hand side, we get 16. Okay? Now let's solve for why I have why minus three squared. And I'm just gonna leave that, uh, X plus four square together is one chunks. We move that over, I'm going to take the square root that typically when we take a square root, it is a plus or minus. However, I know that I only have half my circle. The pluses, the upper half of the circle, the minuses. The bottom half. Since I Onley have the bottom, that's going to be Ah plus. So I could erase that plus or minus and just put a plus in. And now I'm going toe. Add three to both sides, add three and remove it from the right hand side or left hand side there. So that's my function. This is F of X. Now, now we know our function. Let's do some evaluating. I'm going to pick three points that I want to evaluate. Uh, at this function at I would have f of negative three f of negative six and f of negative one. Okay, so let's take a look first. Let's put in negative three. This is going to give me three plus. Well, that's gonna be 16 minus negative. Three plus fours is 11 squared is one. So this is going to be three plus square root of 15. Now, how about Rx equaling negative six. Well, again, I have three plus the square root of 16, but in this case, negative six plus four is negative. Two squared is four. So this is going to be three plus the square root of 12. And I can simplify that a bit to to square root of three for that for that radical last one. If f f f of X Um, if I'm putting in for negative one for X, that's going to give me three plus the square root of 16 now negative one plus four is going to be three. That's going to give me a nine, which gives me three plus the square root of seven. So those are my three answers. The three values didn't mean to make that move. Sorry about that. Three values for my function at my three given exes.

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