Splitting up curves The unit circle x² + y² = 1 ...

Question

Splitting up curves The unit circle x² + y² = 1 consists of four one-to-one functions, ƒ₁ (x), ƒ₂(x) , ƒ₃(x), and ƒ₄ (x) (see figure).

b. Find the inverse of each function and write it as y= ƒ⁻¹ (x)

Accepted Answer

Welcome back, everyone. The ellipse shown below is defined by the equation 4 X2 + y2 equals 4. It consists of 41 to 1 functions G1 G2 G3 and G4 of X. What is the inverse function of G4 of X? What is the domain of the inverse function? So first of all, we're looking at Y equals G4 X. Which is the part in the 4th quadrant of the coordinate plane, and looking at this function, what we can tell is that the domain is between 0 and 1, right? Essentially those will be inclusive, so we can say that the domain is between 0 and 1, so 0 is less than or equal to X. And X is less than or equal to 1. Now, what about the range? Well, essentially we can see that the Y values extend from -2 up to 0, right? We can label that from -2 up to 0. We already know that X extends from 0 to 1. So for the range, we can say that it's between -2, which is less than or equal to Y. Andy is less than or equal to 0. Now these will be important later, but now let's focus on the general expression of the function 4 X2 + y2 equals 4. We are solving for the inverse, right? And if we're solving for the inverse, this means that we want to solve for X. Now, solving this equation for X we're going to get or X2 equals or minus Y2 and now dividing both sides by 4, we get X2 equals. Or minus y2 divided by 4. And now if we take square root of both sides, we can simply indicate that we are taking square root of both sides. We're going to get positive or negative X. We call that square root of X where this positive or negative X or the absolute value of X, and then we're going to rewrite the right hand side. We get. 4 divided by 4, which gives us 1 minus y2d. Divided by 4. Now we will need to take either the positive sign or the negative sign, but before we do that, because this is an inverse function, well, we simply need to swap X and Y, right? So we're going to say that positive or negative Y equals square root of 1 minus Y2 divided by 4. Now, we want to understand the range of this function, because based on the range, we will take either the positive or the negative sign, and the range. Of this function. Is the domain. Of the parent function. Or the domain of G4. And because it had a domain between 0 and 1, this means that we're going to take the positive sign, right? Because the domain is defined for positive values including zero. So we can say that G inverse, which is our Y. 4 of X is equal to square root of. 1 minus. X squared divided by 4. So there's a minor mistake. Previously when we swapped the terms, we had to write 1 minus X squared divided by 4. So now everything is consistent, and this will be our inverse function. Now let's consider the domain. For the domain, we have to recall that domain is the range of the parent function. So if we're considering. The inverse function its domain is equivalent to the range of the parent function. So in this case it would be between 2 and 0. We're going to write -2 less than or equal to X because now we're looking at the domain, which is less than or equal to 0. And that's it. We have our final answers. The inverse function G4 of X is equal to square root of 1 minus X2 divided by 4, with a domain of X being between 2 and 0 inclusive. Thank you for watching.

Splitting up curves The unit circle x² + y² = 1 consists of four one-to-one functions, ƒ₁ (x), ƒ₂(x) , ƒ₃(x), and ƒ₄ (x) (see figure)<IMAGE>.

b. Find the inverse of each function and write it as y= ƒ⁻¹ (x)

Key Concepts

Inverse Functions

One-to-One Functions

Unit Circle

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