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A composite function is a function that is created when one function is evaluated inside another function. It's like a mathematical nesting doll, where one function lives inside another. In simpler terms, if you have two functions, say \( f(x) \) and \( g(x) \), and you "plug" \( g(x) \) into \( f(x) \), the resulting function \( f(g(x)) \) is a composite function.
Composite functions are notated as \((f \circ g)(x)\), which stands for \( f(g(x)) \).

• The order of composition is important: \((f \circ g)(x)\) is not the same as \((g \circ f)(x)\).
• Always ensure the inner function \( g(x) \) is fully evaluated before substituting it into \( f(x) \).

An example would be based on our original problem where \( g(x) = \frac{1}{x+3} \) is substituted into \( f(x) = \frac{1}{x} \) to create \( f(g(x)) = \frac{1}{\frac{1}{x+3}} \). Solving this, you'll simplify it to \( x+3 \), demonstrating how composite functions are carried out.

Rational functions are expressions of the form \( \frac{P(x)}{Q(x)} \), where \( P(x) \) and \( Q(x) \) are polynomials and \( Q(x) eq 0 \). These functions are very common in algebra and calculus, and display interesting behaviors through their domains, asymptotes, and intercepts.
In our problem, both \( f(x) = \frac{1}{x} \) and \( g(x) = \frac{1}{x+3} \) are rational functions because they have polynomial expressions in the numerator and denominator.

• A key property of rational functions is their domain, which is defined as all possible values of \( x \) that do not make the denominator zero.
• For \( f(x) \) and \( g(x) \), critical values need to be found where their denominators become zero, making \( f(x) \) undefined at \( x = 0 \) and \( g(x) \) undefined at \( x = -3 \).

These characteristics help us understand how to approach and manipulate rational functions during composition or function operations like in our composite function \( f(g(x)) \).

Function evaluation is simply the process of finding the output of a function for a particular input. This process is fundamental in mathematics because it helps check how functions behave and what outputs they produce for specified inputs.
When evaluating a function, the given input is substituted in place of the variable \( x \) in the function's formula. For instance, if you have \( f(x) = \frac{1}{x} \) and you wish to evaluate \( f \left( \frac{1}{x+3} \right) \), you replace every \( x \) in \( f(x) \) with \( \frac{1}{x+3} \). This results in the evaluation \( \frac{1}{\frac{1}{x+3}} \), which after simplification gives \( x + 3 \).

• Always simplify the expression as much as possible after plugging in your inputs to get the final result.
• Look out for restrictions or undefined values linked to the input values substituting \( x \).

Through function evaluation, we thoroughly determine how inputs are mapped to outputs, which is critical for understanding and solving mathematical problems involving functions.