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) = 2 / ( x² + 2)

Accepted Answer

Welcome back everyone. In this problem, we want to identify the inverse functions and their domains. For the function K of X equals five divided by X squared plus five. For our answer choices, A says it is the positive and negative values of the square root of five multiplied by one divided by X minus one where the domain is open parentheses, +01 closed bracket. B says it's the positive and negative values of the square root of five multiplied by one divided by X minus one where the domain is open parentheses. 05 closed bracket. C says it's the positive and negative values of the square root of one divided by X minus one where the domain is open parentheses, 02 closed bracket. And D says it's the square root of the positive and negative values of the square root of one divided by X minus one where the domain is open parentheses, 03 closed bracket. Now, in order to identify the inverse function and their domains, let's make sure we understand what the function K FX is and what its domain is. OK. So now for our function A of X equals five divided by X squared plus five. OK. Then we can tell that the domain of chaotic is all real numbers. OK. And that's true because X squared plus five is always going to be positive and is never going to be zero. OK. So the domain of K of X, let's just highlight that here is all real numbers. So we can keep that in mind. Now, if we're going to identify the inverse function, that means first, we can let Y OK be equal to K of X, OK. And that means our function is not gonna be Y equals five divided by X squared plus five. And then we can swap X four Y OK. And now when we do that, that means we're going to, our equation is going to become X equals five divided by Y squared plus five. We can find the inverse by solving for a while. Now solving for Y OK. Let me make some space here from our equation. If we multiply both sides by Y squared plus five, then five multiplied by Y squared plus five, sorry XX multiplied by Y squared plus five is going to be equal to five. That means Y squared plus five equals five divided by X Y squared is going to be equal to five divided by X minus five if we subtract five from both sides. And no, that means Y OK. If we square root both sides, Y is going to be the positive and or negative value of the square root of five divided by X minus five. And now, in this case, we can factor out five from what's inside our square root. So we can say Y equals plus or minus the square root of five multiplied by one divided by X minus one. That means then that our inverse function K inverse of X because remember we said when we solve for Y that would be the inverse function is going to be plus or minus or expression. In other words, it's going to be the positive value of the square root of five multiplied by one divided by X minus one or I say and OK, it's going to be the negative value of the square root of five multiplied by one divided by X minus one. So now we have our inverse function, the two possible values. What is our domain going to be? Well to find the domain. OK. Did the men of our inverse function? Yeah. Is the range is going to be the range of our original function. So if we find the range of K of X, that will be the domain of K of the K of inverse of X. OK. Now how can we do that? Well, OK, since X squared is always positive, OK, then that means K FX has a maximum value. OK. K FX will be at its maximum when X equals zero. OK. So now to find the maximum value of K FX we can solve for K of zero. So that means then K of zero is going to be equal to five divided by zero squared plus 5, 0, squared plus five equals five and five divided by five equals one. OK. So we know here that the maximum value of K FX is one and as X approaches infinity, OK. In other words, as X gets larger, then K of X approaches zero. And that makes sense because X is an R denominator. And if the larger our denominator gets, the smaller our result. In other words, the closer it gets to zero. Therefore, that tells us then OK, the range of care effect, OK is going to be open parenthesis, 0 to 1 closed bracket. And remember we said that the range of K effect is the same as the domain of the inverse. Therefore, the domain of the inverse of K is going to be 01, sorry, open parenthesis, 01, close the bracket. Thus, that would be our inverse function. OK. Those are, in other words, we have the values for our inverse function and we've also figured out its domain. 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) = 2 / ( x² + 2)

Key Concepts

Inverse Functions

Domain and Range

Finding Inverses Algebraically

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