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), ….

y² = x²(4 − x) / 4 + x (right strophoid)

Accepted Answer

Welcome back, everyone. Determine the functions Y equals F1 of X, Y equals F2 of X, and so on that are implicitly defined by the equation Y2 equals 3x2 multiplied by 5 minus x divided by 6 + 2 X. For this problem we want to define functions Y equals F of X, and we're given the Y2 equals. 3 X squad multiplied by 5 minus X divided by 6 + 2 X. We can notice that we're only given a variable Y on the left. And on the right, we're given Xes only, right? So there's a function of X. But now what we want to do is solve for Y because, well, we're given the Y squared, right, and we need Y. So what we're going to do is simply take square root of both sides. So we're going to get square root of Y2d. It's going to be equal to square root of 3 x 2 multiplied by 5 minus X divided by 6 + 2 X. We have to recall that square root of Y2 is the absolute value of Y, right? Or basically, Plus or minus Y and this is equal to square root of the right hand side. Now if we want to simplify, well, we can take square root of X2, which is also the absolute value of X, right? So we're going to get the absolute value of X multiplied by. Square root of 3 multiplied by 5 minus x divided by 6 + 2 X. And now if we simply consider the positive or negative sign, we can just move it in front of the absolute value of X. And we're going to get Y1. This is our first function, right? We can define it as the positive one, right? So we can just use the positive sign and we're going to get the absolute value of X multiplied by square root of 3. Multiplied by 5 minus X. Divided by 6 + 2 X. Now the second function, which is going to be Y2 is going to be the function with a negative sign. So we want to write Y2, our second solution for Y, and it is going to have a negative sign. So now we have the two possible functions for Y, and we're going to label them as our final answer. Thank you for watching.

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), ….y² = x²(4 − x) / 4 + x (right strophoid)

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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), ….
y² = x²(4 − x) / 4 + x (right strophoid)