Finding all inverses Find all the inverses associated with the...

Question

Finding all inverses Find all the inverses associated with the following functions, and state their domains.

ƒ(x) = x² -2x + 6

Accepted Answer

Welcome back everyone. In this problem, we want to provide all the inverses of the function G of X equals X squared plus four, X plus seven and identify their domains. A says there are negative two plus the square root of X minus three and the negative two minus the square root of X minus three where the domains are X is greater than are equal to three. B says it's negative two plus the square root of X minus seven. And the negative two minus the square root of X minus seven where their domains are X is greater than or equal to seven, C negative two plus the square root of X minus five. And the negative two minus the square root of X minus five where their domains are X is greater than or equal to five and D negative two plus the square root of X minus one and negative two minus the square root of X minus one where their domains are X is greater than or equal to one. Now let's first try to figure out the inverse of the function and then we'll think about their domains based on the result. OK. So, so far we know that our function G FX is equal to X squared plus four X plus seven. OK. Now we can start to find the inverse by replacing G of X with Y, OK. And when we replace G FX with Y, then our function reads Y equals X squared plus four X plus seven. If we subtract seven from both sides, Y minus seven equals X squared plus four X and no, our equation is perfectly set up for us to solve by completing the square. OK. Now, by completing the square here, notice that from our problem, we can tell that a, the coefficient of X squared equals one B, the coefficient of X equals four. And C, OK, we know that C to, to complete our square, the constant, we need to complete our square is going to be equal to half of B squared. So that's a half of four squared, that's two squared, which equals four. So to complete the square, we can add four to both sides of the equation. So now our equation is gonna be Y minus seven plus four equals X squared plus four, X plus four and no, X squared plus four, X plus four is equal to X plus two, R squared. While Y minus seven plus four is Y minus three. Now we can swap X and Y, OK. And when we swap X and Y, then our equation now becomes X minus three equals Y plus two squared, we can find the inverse function by solving for Y OK. Because now to solve for Y, now I can put this in red here to solve for Y. First of all, we can square root both sides OK. And then we're going to get Y plus two being equal to the positive and negative square root of X minus three. Therefore, this tells us that Y is going to be equal to negative two plus R minus the square root of X minus three. OK? No, no, that we've solved for Y that means we can replace Y with our inverse. So now this tells us then that the inverse function of X, OK must be equal to negative two plus R minus the square root of X minus three plus R minus tells us that there are two possible values. So this means then that the inverse function is either going to be equal to negative two plus the square root of X minus three or OK, it's going to be equal to negative to minus the square root of X minus three. So we know the possible solutions for or inverse values. Next, let's try to figure out their domains. What are they going to be? Well, for our domain for our function to work, we know the expression inside the square root must be nonne OK. So since the expression inside the square root, OK must be non negative. This tells us then that X minus three must be greater than, or equal to zero, which means then that X must be greater than, or equal to three. And since both of our expressions are under the square root, OK. Both when it is negative two plus square root of X minus three and negative two minus the square root of X minus three. That means the domain of both of our functions must be X is greater than or equal to three. Therefore, that tells us then that A is the correct answer. Thanks a lot for watching everyone. I hope this video helped.

Finding all inverses Find all the inverses associated with the following functions, and state their domains.

ƒ(x) = x² -2x + 6

Key Concepts

Inverse Functions

Domain and Range

Completing the Square

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