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Determine (a) $f(x)$ ) and the domain of the composite function, (b) $g(f(x))$ and the domain of the composite function.$f(x)=2 x+3 \quad g(x)=x^{2}+3$

(a) $2 x^{2}+9-\infty<x<\infty$(b) $4 x^{2}+12 x+12-\infty<x<\infty$

Algebra

Chapter 1

Functions and their Applications

Section 2

Basic Notions of Functions

Functions

Campbell University

Idaho State University

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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for this problem. We've been given two equations F of X equals two X plus three and G f X equals X squared plus three. And our goal is to make two composite functions F of G of X, n g of F of X. We're gonna find these composite functions and make sure we have the proper domain for each of them. So how do we put together a composite function? Well, remember, when we do a function like F of X if I do something, If I write, this is f of three. It means that I'm gonna go to my F function. And anywhere there is an ex I'm gonna put a three. Well, for a composite function, I have f of G of X. So I'm going to my F function, which is two X plus three. I'm going to take out the X, and when I'm putting in is my GI function X squared plus three three outermost function. This gets this case F tells me the function I'm going to first. The other one is the one I'm plugging in. So let's get rid of some of our parentheses here. This gives me two X squared plus six, and I already have a plus three. So that's two X squared plus nine. Now let's check out the domain I can't have. I have to make sure that the numbers go are in the domain of G and in the domain of I full composite function f of G of X. Well, neither G nor the composite function have anything that would invalidate a nen. Put every input will work. So my by a domain is all real numbers from negative infinity to infinity. Okay, now, what about G of F of X? Well, now the outside function is my GI function, so that's where I'm going first. My GI function says I have X squared plus three. I'm just kind of leaving a blank. Therefore, I'm saying, X, I'm going to take out the X. What I'm putting in is F of X, which is two X plus three. So let's multiply that out. Two X plus three squared is going to be four X squared plus 12 X plus nine and already have a plus three. So that's actually going to be a plus 12. So that's my composite function. G of F of X again for the domain. Anything in the domain has to be in the domain of ffx because we're doing that. We kind of evaluating that piece first as that innermost piece and it has to be in the domain of my entire composite function. So in this case, there are no numbers that are excluded. So my domain is all real numbers from negative infinity to infinity.

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