Find the inverse

Question

Find the inverse $f^{-1}\left(x\right)$ of each function (on the given interval, if specified).

$f\left(x\right)=\frac{2}{x^2+1}$ , for $x\geq{0}$

Accepted Answer

Welcome back, everyone. Find the inverse function G inverse of X of G of X equals three divided by X squared plus four or X greater than, or equal to zero. We're given four answer choices A B and C as well as D and all of these, they have square root of a function, right? And under a root we have different expressions. So we want to essentially understand which one is the correct answer. So first of all, we're going to define the function as the problem suggests, G of X equals three divided by X squared plus four, we're also given the domain of the function X is greater than, or equal to zero. So basically, it's non negative from here. What we're going to do is basically use a technique in which we solve for the inverse function. As follows. We're going to rewrite G of XY, right? That's our Y coordinate and it is equal to three divided by X squared plus four. Our next step is to switch X and Y. So now instead of Y, we're going to have X and instead of X, we're going to have Y, now we want to solve for Y. So first of all, we have to divide both, we have to divide three by X to get our denominator. So three divided by X, would it be equal to Y squared plus four? We are now going to subtract four from both sides. So three divided by X minus four would be equal to Y square. So Y squared is equal to three divided by X minus four. Meaning if we want to identify the value of Y, we have to take square root of both sides. So we're taking square root of the left hand side in square root of Y squared, square root of Y squared essentially gives us the absolute value of Y which is simply positive or negative off the left hand side. So square root of we divide by X minus four and we have to choose one of them. Our function must be monotonically increasing or decreasing in its range will be equal to the domain of the parent function. So if X was non negative, now Y must be non negative, we're going to say that range would be Y greater than or equal to zero, looking at the original domain. So we're going to choose the positive value and we can see that why, which is G inverse would be equal to square root of three divided by X minus four, which essentially corresponds to the entero C. And that's our final answer to this problem. Let's label it and conclude the video. Thank you for watching.

Find the inverse $f^{-1}\(\left\)(x\(\right\))$ of each function (on the given interval, if specified).
$f\(\left\)(x\(\right\))=\(\frac{2}{x^2+1}\)$ , for $x\(\geq{0}\)$

Key Concepts

Inverse Functions

Finding Inverses Algebraically

Domain and Range

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