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.

c. y = x¹/⁴

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, and m of X, which is equal to x + 2, and n of X, which is equal to 4 x 2. Express Y equal to 4 x as a composition involving one or more of the given functions. We're given 4 possible choices as our answers. For choice A, we have N of L of X. For choice B, we have N of L of M of X. For choice C, we have N of L of K of X, and for choice D we have N of M of L of X. Now, we need to express our function Y, which is equal to 4X, as a composition involving one or more of the given functions in the problem. So, what we want to do is determine a starting point. Since our expression that we're looking for is Y is equal to 4 X, we're gonna look for a function that has a factor of 4 in it. And the only function that has a factor of 4 in it is N of X. We're going to use that as our starting point. So, we need to determine what the actual input is for a function in of X. So, we'll have N of F of X. And that's gonna be equal to. 4 multiplied by the quantity of F of X. In quantity squared, and this is going to have to be equal to our Y value, which is 4 X. So what we want to do now is solve this for F of X. So, I can start off by dividing both sides of the equation by 4, and so then I'll have the quantity of F of X. In quantity squared is equal to X, and taking the square root of both both sides will have F of X. is equal to the square root of X. So here we've dropped the negative solution for the square root of X. Since we want our Y value to be a positive 4X, so we're looking for the positive root only, so we can ignore the negative root. So, if I look at our given functions, none of our functions are equal to the square root of X, and there's actually only one function that has a square root in it, and that is our function, L of X. So, we need to do now is figure out the input for L of X that will give us the square root of X out. So, we'll look at of X, or sorry, L of G of X. And this is going to be equal to. The square root of the quantity of G of X. Minus 2 in quantity, and this has to be equal to F of X, which is the square root of X. So we're going to do now is solve this for G of X, so squaring both sides, we'll have G of X. Minus 2 is equal to X, and then we just need to add 2 to both sides, and so we'll have G of X. is equal to x + 2. And if we look at the functions that we're given, this actually corresponds to M of X. So this is going to be equal to M of X. So now we're able to actually write out our final answer. So that means we can write Y as being equal to N L M X. And so that is actually 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 actually corresponds to answer choice B. 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.

c. y = x¹/⁴

Key Concepts

Composition of Functions

Function Notation

Inverse Operations

Watch next