Composition of Functions

Let...

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.

a. y = 2x − 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 2X minus 4, L of X, which is equal to X + 6, and M of X, which is equal to 4 X. Express Y is equal to 8 X + 8 as a composition involving the given functions. We have 4 possible choices as our answers. For choice A, we have K M L X. For choice B, we have K of M of M of X. For choice C, we have M of K L X, and for choice D, we have M of L of K X. Now we need to express Y, which is equal to 8 X + 8 as a composition of the given functions in the problem. Now, if we look at our function for Y, It is equal to 8 X plus 8. So, we're actually going to start with our function M of X, which is equal to 4X, since we need an X variable in our final answer. So, what we want to do is figure out the input for M of X. So we'll call that input F of X. So, we're going to have M of F of X. And this is going to be equal to 4 multiplied by F of X. And this needs to be equal to our expression for Y, which is equal to 8 X plus 8. So, we can solve this expression for F of X by dividing both sides of the equation by 4, and we'll get F of X. is equal to 2 X plus 2. So now we need to find an input or one of the other functions that we were given that would give us our function F of X out. Now if we look at the other given functions, we see that our function LX has an addition between the linear term and the constant term, which is what we want for F of X. We're going to use that as our starting point. So we're gonna call the input for L of X G of X, we'll have L of G of X. And this is going to be equal to G of X plus 6. Which needs to be equal to F of X, which is equal to 2 X plus 2. And so we can solve this for GFX by subtracting 6 from both sides of the equation. In doing so, we'll get GFX. It's equal to 2 X minus 4. And if we look at the other functions that we are given in the problem, this actually corresponds to K of X. So that means that G of X is equal to K of X. So we now have enough information to write our final answer. So that means Y is going to be equal to M L K X. So this here is going to be the final answer for our problem. And if we can compare our answers to our answer choices, we see that the correct answer choice corresponds to answer choice D. 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.

a. y = 2x − 3

Key Concepts

Composition of Functions

Linear Functions

Function Transformation

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