Inverse of composite functions

b. Let g(x) =...

Question

Inverse of composite functions

b. Let g(x) = x² + 1 and h(x) = √x. Consider the composite function ƒ(x) = g(h(x)). Find ƒ⁻¹ directly and then express it in terms of g⁻¹ and h⁻¹

Accepted Answer

Welcome back, everyone. Find the inverse F inverse of the composite function F of X equals G H X directly and also represent in terms of G inverse and H inverse, where G of X is equal to x + 2 and H of X is equal to x to the power of 1/3. So let's identify the given functions we're given a G of X equals x cubed plus 2. And the problem says that H of X is equal to X to the power of 1/3. We want to find the inverse of F knowing that F of X. Is G of H of X. So let's first all find F of X, and basically we have the outer function, which is G and the inner function which is H. So we have to substitute the inner function into the outer function, and this gives us X cubed, which is basically H cubed now plus 2. So we get X to the power of 1/3 cubed. Plus 2. Now, let's raise X to the power of 1/3 to the power of 3rd. This gives us X, and then we have X + 2. We have found the function F of X, and then we can find the inverse. To find the inverse, we are simply going to rewrite it as Y equals x + 2. Swap X and Y, which gives us X equals Y + 2, and solve for Y by subtracting 2 from both sides. So Y equals x minus 2. Now we have one of the required answers we can say that the inverse itself. In verse I. Of X is equal to x minus 2. And now we want to express it in terms of G inverse and H inverse, and this means that we have to identify each. Let's begin with G inverse. We know that Y equals x + 2. We swap X and Y such that X becomes equal to YQ + 2. We subtract tea from both sides, and this gives us. Y cubed equals x minus 2, and if we want to solve for Y, we basically raise each side to the power of 1/3, which gives us Y equals. X minus T. It's the power of 1/3. And that's G inverse of X. We can also identify H inverse knowing that H is equal to X to the power of 1/3, so we can say that Y equals. X is the power of 1/3 and we swap X and Y. So x equals twice the power of 1/3. And if you want so for Y, well, we essentially raise both sides to the power of 3. So we get Y equals x cubed. So that each inverse of X is equal to x cube. Now, if we carefully analyze each result, let's look at the G inverse. Let's look at the H inverse, and now let's look at the F inverse. Well, we are simply taking X minus 2 to the power of 1/3 and raising it to the power of 3, right? So essentially we can show that if we take X minus 2. To the power of 1/3 and then raise it to the power of 3, we're going to get X minus 2. Now, the inner function is the inverse of G and the outer function is the cube, right? So we can say that the inverse of H at the inverse of G of X. Gives us the inverse of F. Which essentially means that we have successfully. Identify the form in terms of the two inverses. Let's label our answers. So we have the expression for the inverse of F in terms of X, and we have the same expression in terms of inverses of the functions H of X and G of X. Thank you for watching.

Inverse of composite functions

b. Let g(x) = x² + 1 and h(x) = √x. Consider the composite function ƒ(x) = g(h(x)). Find ƒ⁻¹ directly and then express it in terms of g⁻¹ and h⁻¹

Key Concepts

Composite Functions

Inverse Functions

Function Notation and Operations

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