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Let $y=g(x)$ describe the lower half of the circle $x^{2}+y^{2}=16 .$ Determine (a) $g(-2),$ (b) $g(0),$ (c) $g(2)$

(a) $-2 \sqrt{3}$(b) -4$(c)-2 \sqrt{3}$

Algebra

Chapter 1

Functions and their Applications

Section 5

The Circle

Functions

Oregon State University

Harvey Mudd College

University of Michigan - Ann Arbor

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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04:30

Let $y=g(x)$ describe the…

05:28

Let $y=f(x)$ describe the …

02:29

01:28

If the circle $x^{2}+y^{2}…

04:35

03:26

Find equations for the upp…

02:14

Graph the lower half of th…

00:32

Sketch the circle. Identif…

01:57

Find the center and the ra…

to this problem. We have been given a circle X squared plus y squared equals 16. Let's take a moment and just sketch What this looks like This is gonna be a pretty rough sketch. Um, since it equal 16, that means my radius is the square root of 16 or four. So I'm going four units all the way around and my center is going to be a zero because I have just an X squared in a Y squared. I've not X plus anything or why, minus anything that puts it centered at the origin. Now, the problem with this a circle is not a function. It fails the vertical line testify, draw a line through It's going to hit the circle in two points. So what we've done here is we've modified it. We're only going to be looking at the lower half of the circle, and I'm just gonna trace that piece here in red. Now, as long as I only do the lower half of the circle that peace and red is a function. There's Onley one output for every input so I can define my function. Oh, and I copied it wrong. I copied it His f I'm just so used to that this is actually a GI function. Okay, so I know now that I'm going to have a function. Let's solve our circle equation for why, So I can see what my function actually is. So I have y squared equals 16 minus X squared, and then I take the square root. Now, usually when we take the square root, we would say it's plus or minus the square root of that number. But remember, I Onley want the bottom piece. The positive square root is the upper half of the circle. I want the lower half so I don't have to do plus or minus. I know it's just going to be negative. I want every Y value to be a negative value for this function. Okay, so there is my function. This is equal to G of X, and I have three numbers that we're going to evaluate. I want to know the value of my function at negative two at zero and at positive two. So let's plug these in if X is negative. Two. That means I have the opposite or negative square root of 16 minus four negative square root of 12. And we can simplify this. There's a factor of four under that radical. So I could say that this is negative Two square root of three. And I'm actually going to skip to the bottom of these three. If X is positive too, it actually makes no difference. It's still going to be 16 minus four under that radical. So this is also going to be negative to square root of three. And actually, that kind of makes sense. If you look at the circle, if I have a value of X equals negative to our positive too, they should have the same y value. Okay, Last one. If X is zero, that gives me the negative square root of 16 minus zero, which is going to be negative four. So these are my three outputs for my function at my three given X values

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