for the given functions f and g, find: (a) f ∘ g (b) g ∘ f (c) f ∘ f (d) g ∘ g

step 1 All right, we're composing two functions together, and F of G. We'll start that off. That'll be

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

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

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Key Concepts

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.

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.

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