In Problems 23-38, for the given functions f and g, find: (a) f ∘g (b) g ∘f (c) f ∘f (d) g ∘g St

step 1 F of x equals x over x 3, and G of x equals 2 over x. So our limitation in F of x, we have x suc

In Problems 23-38, for the given functions f and g, find:...

In Problems $23-38,$ for the given functions $f$ and $g,$ find:
$(a) f \circ g$
(b) $g \circ f$
(c) $f \circ f$
(d) $g \circ g$ State the domain of each composite function.
$$
f(x)=\frac{x}{x+3} ; g(x)=\frac{2}{x}
$$

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Key Concepts

Function Composition

Function composition is the process of applying one function to the results of another. In notation, (f ? g)(x) means f(g(x)), which implies that you substitute the output of g(x) into the function f. This concept is fundamental for combining functions and understanding how transformations of inputs work through successive operations.

Domain of Composite Functions

The domain of a composite function consists of all values of x that can be input into the inner function such that its output is within the domain of the outer function. This requires checking the restrictions from both functions, especially when they involve denominators or other expressions that cannot take certain values, to ensure that the composite function is well-defined.

Rational Functions

Rational functions are functions that can be expressed as the ratio of two polynomials. They often include restrictions on their domains due to the possibility of division by zero. When composing rational functions, it is important to simplify the expressions and carefully analyze the values of x that cause any denominator to become zero in either the individual functions or their compositions.

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