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(a) Determine if the given equation is a supply or demand equation. For a supply equation, find the minimum price for which there will be any supply. For a demand equation, find the maximum possible demand and the maximum price that can be charged. (b) Plot the graph of the given equation.$$3 p+6 x^{2}=12$$

(a) Demand $x=\sqrt{2}, p=4$

Algebra

Chapter 1

Functions and their Applications

Section 6

Economic Functions

Functions

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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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given the function three P plus six X squared is equal to 12. We first want to determine whether or not this is supply or demand function. So to do that, let's start by solving for P. We can take three P. We're gonna subtract six X squared from both sides to get three P. Is equal to 12 minus six X squared, Dividing both sides by three. We have P is equal to 4 -2 x squared. Now to know whether or not this is a supply or demand curve. We want to look at the sign of our slope. We can see that it's negative because it's negative. This has to be a demand function. Considering that demand curves will always have that negative slope. Supply curves, love a positive slope. So we can start graphing this a little bit if we want. We know it's a demand curve which is obviously downward sloping. And because we were working with a non linear function, you can see that we have this squared term. That means that our curve is actually going to be non linear like I said. So it is actually going to be a curve rather than a straight line. So it might look something like this. And now what were we want to do next is find our maximum price and demand. So let's start with finding this maximum price. And we can see graphically is that that that highest price on this curve occurs right here at this point. And we can see that that's happening when our quantity is equal to zero and our quantity is represented as X. In our function. So we need to do is plug in zero for X. So we'll take our demand function where P is equal to four minus two X. Squared. But we have this remember we're plugging zero in for X. You can see that whole term cancels out which gives us a price equal to four And that might be $4. I don't know what units were working in but will say that it's $4 is our maximum price. Now our maximum demand on the other hand, so that's when our quantity is as high as it gets. That occurs down here. You can see that that's the highest quantity and that's happening when our price is equal to zero. So we're going to do the same thing that we just did accept plug in zero for price. So we'll have zero is equal to 4 -2 x squared. Now we just need to solve for X. So I'm going to move the two X squared over to the left hand side. So we have two X squared is pulled A four divide both sides by two gives us X squared is equal to two, and then we just need to take that square root, and we can see that X is equal to the square root of two, and that would be our maximum quantity. So if we wanted to make our graph a little bit more detailed, we could add This route to over here and four over here.

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