The following equations implicitly define one or more function...

Question

The following equations implicitly define one or more functions.

b. Solve the given equation for y to identify the implicitly defined functions y=f₁(x), y = f₂(x), ….

x+y³−xy=1 (Hint: Rewrite as y³−1=xy−x and then factor both sides.)

Accepted Answer

Hello. In this video, we are going to be determining the functions of Y that are implicitly defined by the equation 4 X plus Y cubed minus 2 XY equal to 8. We are going to rearrange the terms of the equation as follows, and then factor both sides. So, the problem wants us to start with this form of the equation. Y cubed minus 8 is equal to 2 XY minus 4 X. And we want to use this equation to implicitly find the equations of Y. Now, let's start by factoring both sides of the equation. Y cubed minus 8 can be factored using a difference of cubed terms. That is going to give us Y minus 2 multiplied by the quantity, Y2 + 2 Y + 4. On the right hand side, we can factor out a 2 X term, and that will allow us to rewrite the right hand side of the equation as 2 X multiplied by the quantity Y minus 2. Now, in this step of the problem, we can divide both sides of the equation by Y minus 2, but there are two interesting characteristics about this, about this step. First, we can only divide by Y minus 2 as long as Y minus 2 is not equal to 0, but we also have to consider the possibility that Y minus 2 is equal to 0. So, what we are going to have to do is we are going to have to check for the possible functions of Y when Y minus 2 is not equal to 0, and when Y minus 2 is equal to 0. Let's go ahead and start by checking for the function of Y, when Y minus 2 is equal to 0. Now, when this function is equal to 0, all we have to do is just solve for Y. We can do this by adding 2 to both sides of the equation, and that will give us Y is equal to 2. What this means is that the first explicit function for Y will be F1. Of, I'm sorry, of X equal to 2. This is going to be the first function of Y. Now what about the other half when Y minus 2? is not equal to 0. Well, when Y minus 2 is not equal to 0, we can simplify the function. The equation will simplify to Y2 + 2 Y + 4 equal to 2 X. What we are going to do is we are going to take this equation, set it equal to 0, and solve for the functions of Y. In order to do this, we will subtract both sides of this equation by 2 X. We will have Y2 + 2Y + 4 minus 2X equal to 0. In this step, we are going to group up the constant values that don't include factors of Y. That is going to be 4 minus 2 X. What we can do now is we can solve for Y using the quadratic formula. That is going to give us Y is equal to -2 plus or minus the square root of 22, which is 4. Minus 4 multiplied by the coefficient of the first term, which is 1, multiplied by the constant value, which is 4 minus 2 X, and this will all be divided by 2 multiplied by the coefficient of the first term, which is 1. So we will have a denominator of 2. Now we are going to simplify. -4, multiplied by 1, will be -4, and distributing -4 into 4 minus 2X will give us -16 plus 8X. So we have Y. equal to -2. Plus or minus the square root. Of 4 minus 16 + 8 X. This will further simplify, divided by 2, and this will further simplify to be -2 plus or minus the square root of -12 + 8 X divided by 2. We can reorder the terms in the square root. And we can rewrite the square root as 8 X minus 12. We can factor out of 4 from 8 X minus 12, leaving us with Y equal to -2 plus or minus the square root of 4 multiplied by 2 X minus 3, and this will all be divided by 2. We can simplify the 4 in the square root by pulling it out of the square root. That will simplify to be positive 2. So, we have Y equal to -2, plus or minus 2 multiplied by the square root of 2 X minus 3, all divided by 2. If we factor out a 2 from the numerator and eliminate those constants of 2 from the numerator and denominator, we will have Y is equal to -1 plus or minus the square root of 2 X minus 3. What this gives us is two more explicit functions for Y. That will give us F 2 of X equal to -1 plus the square root of 2 X minus 3, and it will give us F 3 of X equal to -1 minus the square root of 2 X minus 3. And these are going to be the last two explicit functions for Y from the original equation. Now, we have these two functions in combination with the one we saw for earlier, where F 1 of X is equal to 2. And these 3 functions are going to be the solution to this problem. So I hope this video helps you in understanding on how to approach this problem, and I will go ahead and see you all in the next video.

The following equations implicitly define one or more functions.
b. Solve the given equation for y to identify the implicitly defined functions y=f₁(x), y = f₂(x), ….
x+y³−xy=1 (Hint: Rewrite as y³−1=xy−x and then factor both sides.)

Key Concepts

Implicit Functions

Factoring

Solving for y

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