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Understanding how to evaluate a function is a fundamental skill in algebra that can be seen as a process of finding the output of a function for a particular input. When we say 'evaluate a function' like in the exercise example where we are asked to find \(f(5)\), we are essentially looking for the value of the function when the input, or the independent variable, is 5.

To evaluate \(f(x)=3x-4\) at \(x = 5\), we replace every instance of \(x\) in the function expression with 5. Here's the algebraic manipulation broken down: \(f(5) = 3(5) - 4 = 15 - 4 = 11\). So, when we substitute 5 for \(x\), \(f(x)\) simplifies to 11.

It's essential that students pay attention to every step of the function evaluation, making sure the arithmetic operations, whether they involve addition, subtraction, multiplication, or division, are executed in the correct order, following the order of operations rules. When this attention to detail is consistently practiced, the process of evaluating functions becomes intuitive over time.

When we are asked to find something like \(g(f(5))\), what's being asked of us is to compute the value of a function inside another function, often referred to as a 'function of a function' or a 'composite function'. Composite functions are one of the more interesting concepts in algebra because they involve an input passing through two different function processes.

To tackle the part of the exercise 'find \(g(f(5))\)', we need to understand that we're using the output of \(f(5)\) as the new input for \(g(x)\). After finding \(f(5) = 11\), we then substitute 11 into \(g(x)\) and get \(g(11) = 11^2 + 6\). The calculation then follows: \(g(11) = 121 + 6 = 127\).

This chain of operations can be extended for any number of functions, creating longer and more intricate composites. The key to mastering this concept is to process each function one step at a time, always remembering that the output of one function becomes the input to the next.

The key to solving various algebraic problems lies in manipulation – the art of rearranging, combining, and simplifying expressions to make them more understandable or to solve for an unknown. Algebraic manipulation underpins the process of evaluating and combining functions, as illustrated in our exercise.

When we evaluate \(f(5) = 3(5) - 4\), and subsequently find \(g(f(5))\), our ability to correctly manipulate these expressions ensures accurate results. This process may include distributing multiplication over addition, combining like terms, or applying exponent rules.

Understanding algebraic manipulation involves knowing and applying the basic arithmetic operations, properties of equality and inequality, the distributive property, and many other mathematical principles. It is a skill sharpened by practice and careful operation. Ensuring every step in the manipulation is valid avoids common errors and leads to the right answer, just as we did in finding \(g(f(5)) = 127\).