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Function evaluation is the process of finding the output of a function given a specific input. It is a fundamental concept and is often the first step when working with functions. In this problem, we are given a table that helps us evaluate functions at specific points.

When you want to evaluate a function, you simply substitute the input value into the function and find the corresponding output. This is often done using tables of values, graphs, or algebraic expressions.

In our exercise, the table provided lists the output values for different inputs. For example, if we want to evaluate the function \(f\) at \(x = 5\), we look at the row corresponding to \(f(x)\) and the column for \(x = 5\). According to the table, \(f(5) = 6\). This tells us that the function \(f\) assigns the value 6 to the input 5.

Composition of functions involves combining two or more functions in such a way that the output of one function becomes the input of the next. This principle is expressed as \((f \circ g)(x) = f(g(x))\). It represents the application of two functions, where the output of \(g(x)\) becomes the input for \(f\).

In this exercise, we are dealing with the self-composition of a function, \((f \circ f)(5)\). This means we first evaluate \(f(5)\) and then use the result to find \(f\) again. So, we start by finding \(f(5)\), which we determined is 6 using the table. Next, we substitute this value into \(f\) again to find \(f(6)\).

The use of composition is prevalent in mathematics as it simplifies complex problems by breaking them into smaller steps. It's important to remember the sequence of operations, as changing the order can lead to different results.

A table of values is a helpful tool in mathematics that organizes information about functions and their outputs. It allows us to quickly find the value of a function at different input points without needing to calculate each time.

In our example, the table lists two columns: one for the input \(x\) and one for each function's output \(f(x)\) and \(g(x)\). We simply locate the desired \(x\) value in the table, then check the corresponding value for the function we are interested in. This straightforward method is very convenient for evaluating and composing functions.

By using a table of values, we can avoid repetitive calculations and reduce the chances of making errors, especially when dealing with complex functions or multiple compositions. It's a powerful strategy for ensuring accuracy and understanding function behavior at various points.