For the given functions f and g, find: (a) f ∘g (b) g ∘f (c) f ∘f (d) g ∘g State the domain of

For the given functions f and g, find: (a) f ∘g (b) g ∘f...

For the given functions $f$ and $g,$ find:
$$(a) f \circ g$$
$$\text { (b) } g \circ f$$
$$(c) f \circ f$$
$$(d) g \circ g$$
State the domain of each composite function.
$$f(x)=\frac{x}{x+3} ; \quad g(x)=\frac{2}{x}$$

Key Concepts

Rational Functions

Rational functions are functions that can be expressed as the ratio of two polynomials. They are commonly encountered in studies of function behavior because they can have restrictions on their domains, particularly where the denominator equals zero. Understanding rational functions is important when composing functions, as the resulting expressions often involve algebraic fractions that must be simplified and whose domains need to be carefully determined.

Function Composition

Function composition is the process of applying one function to the result of another function. It is denoted as (f ? g)(x) = f(g(x)) and involves substituting the output of one function into another. This concept is central in understanding how complex operations can be built from simpler ones, and it provides insight into the behavior of combined transformations.

Domain Analysis in Composite Functions

When composing functions, it is essential to determine the domain of the composite function. This involves considering the domains of both component functions and ensuring that the outputs of the inner function lie within the domain of the outer function. The domain of the composite function is the set of inputs for which both functions are defined, taking into account any restrictions such as division by zero or invalid operations.

Transcript

00:01 Okay, so here we have f of x equal to x over x plus three, and g of x is equal to two over x.

00:06 And for part a, we want to find f composed g of x.

00:11 So that's equal to f of g of x.

00:15 We are inputting g of x into fx.

00:18 So this is going to be equal to we start with our function f, and we input g of x anytime we see an x in f.

00:26 So we have 2, 2 over x, over, well, over 2 plus, well, over, let's give a step here, so over, over 2 over x, plus 3, which becomes equal to 2 over 2 over 6 all over x, getting common denominator there in the bottom, combining our terms, this is equal to 2 over x, well, times the reciprocal, so times x over 2 plus 3x, which is equal to 2 over 2 plus 3x.

01:19 All right, so there is our composite, f composed g.

01:24 Then for the domain, we have to make sure that g of x is, defined as well as our denominator well that to be true we have to have that x is not equal to um x is not equal to um negative 3 and x is not equal to negative 2 3rd right so we have the domain here as the set of all x such that or x is not equal to oh wait a minute wait a minute we're inputting g we're inputting g we're inputing g so we have to have that g of x is defined.

02:18 So we have to have that not x is equal to negative three.

02:20 X is not equal to zero, right? because g of x is what we're inputting into f.

02:25 So therefore, the domain here is a set of all x such that x is not equal to zero.

02:30 Well, if not such that x, such that x is not equal to zero.

02:36 And that x is not equal to negative two thirds would make two plus three x equal to zero.

02:44 So therefore we have to have that g of x is defined as well as the composite f composed g.

02:51 Okay, so there is our domain...

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