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Function evaluation is a fundamental concept in mathematics that pertains to finding the output of a function for a given input. It follows the basic principle that each input, represented by the variable often referred to as 'x', is paired with exactly one output. To evaluate a function, you would replace the variable in the function's rule with the actual input value, and then perform the indicated operations to find the result.

Let's take a basic function like \(f(x) = x + 2\). Evaluating this function for an input of 5 would involve replacing 'x' with 5: \(f(5) = 5 + 2\), thus \(f(5) = 7\). In the exercise, function evaluation was used to show that for the function \(g(x)\), when we input 2, we get \(g(2) = 2\), when we input 4, we get \(g(4) = 4\), but the evaluation for 3 yields \(g(3) = 0\), making the assertion in question false.

A piecewise function is a type of function that is defined by multiple sub-functions, each of which applies to a certain interval of the domain. Each piece is a part of the function that corresponds with specific conditions or restrictions on the input values. Piecewise functions are often used to describe scenarios that have different outcomes depending on the input.

For example, the function used in the counterexample, \(g(x)\), is a piecewise function defined as \(g(x) = x\) for \($x eq 3\) and \(g(3) = 0\). This means that \(g(x)\) follows one rule when \(x\) is any number other than 3, and another rule applies specifically when \(x\) is equal to 3. The notation and graph for piecewise functions can appear more complex because different rules need to be visually represented for different portions of the domain.

Mathematical reasoning is the process through which mathematicians come to conclusions based on logical deductions, proofs, and examples. Counterexamples play a critical role in mathematical reasoning, as they are often used to disprove general statements or conjectures. If a statement asserts that a condition holds for all members of a set, providing just a single instance where the condition does not hold is enough to invalidate the entire statement.

In the assertion from the exercise, the implication was that a particular property (\(f(2)=2\) and \(f(4)=4\) leading to \(f(3)=3\)) held for all functions. However, through mathematical reasoning and the construction of a counterexample, it was demonstrated that this is not the case. It's important to understand that while many examples can suggest a rule, it takes only one counterexample to break it. This is a powerful aspect of mathematical reasoning that safeguards the integrity and precision of mathematical discourse.