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Show how to "build," by composition, the following function $f(x)=\sqrt{x+\sqrt{x+\sqrt{x+\sqrt{x}}}}$.

$$f(u)=\sqrt{u}, u=x+\sqrt{w}, w=x+ \sqrt{v}, v=x+\sqrt{x}$$

Algebra

Chapter 1

Functions and their Applications

Section 2

Basic Notions of Functions

Functions

McMaster University

Baylor University

University of Michigan - Ann Arbor

Lectures

01:43

In mathematics, a function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output. An example is the function that relates each real number x to its square x^2. The output of a function f corresponding to an input x is denoted by f(x).

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00:46

Expressing a Function as a…

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01:10

Use composition of functio…

00:16

Let $f(x)=\sqrt{x}, g(x)=x…

01:01

? Expressing a Function as…

00:52

01:47

For each function $f(x)$ g…

01:44

Let $f(x)=\sqrt[3]{x-4} .$…

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01:33

03:01

For each function f(x) giv…

okay for this problem. We have been given a function f of X, and our goal is to build it back up by compositions. Instead of having just one single function, we want to build it up using multiple functions. Okay, so this is our goal, and we're going to rewrite it as a Siris of a composite function. So we're gonna make g b r outermost function. So let's define them. Are outermost Piece of this function is the square root, because that's this overarching square root sign right there. So we'll say this is square root of something. G of X Are square root of X. Well, what is X? Let's call that you and that is everything under that radical, right? So there's my you. So what I have really is g of you. Okay, well, but let's simplify you a bit. This is X plus again. I have an outermost piece here square roots. So let's call this the square root of something. Um, let's call it W. We'll make a new function W of X. So W is what's under the radical That's X plus the square root of X plus, the square root of X Okay, well, let's do the same thing, because again, I have a square root here. So let's say this is X plus the square root of something. We'll call it V. Okay, but what is V of X v of X is X plus the square root of X, and I don't have those nests cruise anymore, so this might be a good place to stop. So what I really have here is g of you of X because watch what happens. OK, G is my outer outermost function. So if I call the GI function that is the square root of something. Well, what does the square root off you of? X. So you have X is right here X plus the square root of w. Well, what's W. W is X plus the square root of V what's V X plus the square root of X, and that does equal what we were trying to start with. So if I define G of X as the square root of X and then I've got you of X w of X and V of X, I could put all of those together as g of you of X and get back to the original function I started with

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