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Determine the domain of the composite function defined by $y=g(f(x))$.

$\{x | x \text { is in the domain of } f \text { and } f(x) \text { is in the domain of } g\}$

Algebra

Chapter 1

Functions and their Applications

Section 2

Basic Notions of Functions

Functions

Missouri State University

Harvey Mudd College

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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00:44

Find the domain of the com…

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find the domain of the com…

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for this problem. We have a composite function g of f of X. Now what is a composite function? Let's let's let's start there. Ah, composite function is when we put two functions together such that the output of one function is the input of the next. So in this case, F of X is my innermost function. So that means that f of I'm going to evaluate f of X and whatever I get from there that is gonna get input into my G function. So when I'm looking at the domain, I actually have two different things I have to evaluate because I have two functions here. Okay, so first, let's go inside those innermost parentheses. I have f of X, so whatever X I use has to satisfy that function first. So X has to be in the domain of ffx. So whatever that function f of X is, I have to make sure that I'm using a valid X. So no dividing by zero, no square roots of negative numbers, that sort of thing. But now I also have to look at my entire function because I have to have valid just because I have a value coming out of ffx. Maybe I can't put that into G of X. You know, if G is a square root F of X gives me a perfectly valid negative number as a result. But I can't put that into G if it's a square root. Maybe F of X is a perfectly valid number. Going into gives me a zero. It's perfect. But if G e is a division and I'm dividing by X, I can't have a zero. So I have to check. Not Onley. That X is in the domain of EC of ffx, but I need to make sure what I'm getting out of ffx that has to be in the domain of my outermost function. G of X. If both of those were true, that X is in the domain off my composite function.

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