Composition of Functions

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Question

Composition of Functions

Let f(x) = x − 3, g(x) = √x, h(x) = x³, and j(x) = 2x. Express each of the functions in Exercises 11 and 12 as a composition involving one or more of f, g, h, and j.

f. y = √(x³ − 3)

Accepted Answer

Hi, everyone. Let's take a look at this practice problem. This problem says consider the functions K of X, which is equal to x minus 2, L of X, which is equal to the square root of the quantity of X minus 2 in quantity, M of X, which is equal to x + 2, and of X, which is equal to 4 x 2d. Express Y equal to 2 multiplied by the square root of the quantity of X2 minus 1 in quantity. As a composition involving one or more of the given functions. We are given 4 possible choices as our answers. For choice A, we have L of K of N of X. For choice B, we have L of N of KX. For choice C, we have L of M of N of X, and for choice D, we have L of K of M of X. Now we need to express our function Y as a composition involving one or more of the given functions in the problem. And if we look at our function of Y, we see that it involves a square root. So, in order to find a good starting point, we need to look for our functions that we're given in the problem that contain a square root, and there's only one, which is L of X. So, we need to do is figure out the input for L of X that would give us our function Y out. So, we're gonna calculate L of F of X. And that's going to be equal to the square root of the quantity of F of X. Minus 2 in quantity, and this has to be equal to Ry expression, which is 2 multiplied by the square root of the quantity of X2 minus 1 in quantity. So now what we need to do is solve this expression for F of X. So to start, we'll just square both sides of the equation. So we'll have F of X minus 2 is equal to 4, multiplying the quantity of X squad minus 1 in quantity. We'll now distribute the 4, so we'll have F of X minus 2 is equal to 4 x 2. -4, and to solve for F of X, we'll just add 2 to both sides. And so we'll have F of X. is equal to 4 x 2. -2. And if we compare this expression to the other functions that we're given in the problem, none of them exactly match up. But if we look at K of X, we see that we have something that is of similar form. Here we have X minus 2. So we need to do is figure out the input for KFX that will give us our function F of X as this output. So we'll calculate K of G of X. And this is going to be equal to G of X. -2, and this has to be equal to F of X, which is 4 X2 minus 2. Now, to solve for G of X, we just need to add 2 to both sides, and so we'll have G of X. Which is equal to 4 x 2. And if we compare this expression to the functions that we were given in the problem, we see that this actually corresponds to N of X. That means G of X is actually equal to N of X. So we now have enough information to write out our final answer. That means that Y is going to be equal to L of K N X. So this here is the answer that we were looking for for this problem. And if we compare our answer to our answer choices, we see that the correct answer choice corresponds to answer choice A. So I hope that this explanation has been useful, and I'll see you in the next video.

Composition of Functions

Let f(x) = x − 3, g(x) = √x, h(x) = x³, and j(x) = 2x. Express each of the functions in Exercises 11 and 12 as a composition involving one or more of f, g, h, and j.

f. y = √(x³ − 3)

Key Concepts

Composition of Functions

Function Notation

Algebraic Manipulation

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