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When we talk about function evaluation in algebra, we're looking at how to determine the output of a function based on a given input. It's like plugging a value into a formula and seeing what comes out. In the given example, evaluating the function means computing the values of functions like \( g(x) = 3x - 5 \) when \( x \) is a specific number, in this case, 1.

To evaluate \( g(1) \), we simply replace the \( x \) in \( g(x) \) with 1, performing the operation \( 3 \times 1 - 5 \) to get the result, which is \( -2 \). The process seems straightforward, but making small errors in substitution or arithmetic can lead to incorrect answers. Thus, it’s vital to work through each step methodically, ensuring that every part of the function’s operation is applied correctly.

The substitution method is a core part of evaluating composite functions - that's when you have one function inside another, like \( f(g(x)) \). As seen in the original exercise, once we determined that \( g(1) = -2 \), we took that output and used it as the input for another function, \( f(x) \).

Substitution method can be thought of as a 'plug-and-play' approach. You first evaluate the innermost function and then 'plug' its result into the next function. This stepwise approach breaks down complex problems into more manageable ones. The key here is to proceed one step at a time to avoid mixing up values and operations, ensuring that you substitute the correct values correctly into the subsequent expressions. Practicing this method will make it seem less intimidating and more like a simple task of following the path the functions lay out for you.

Algebraic functions are like a special set of instructions that tell us how to manipulate numbers or expressions. They're built using algebraic operations like addition, subtraction, multiplication, division, and exponentiation. The functions \( f(x) = x^2 - x + 4 \) and \( g(x) = 3x - 5 \), as mentioned in our exercise, are classic examples.

These functions act as a rule or a formula in which you put an input (in this case, a number for \( x \)) and get an output based on that formula. For example, our function \( f(x) \) tells us that for any \( x \) we choose, we should square it, subtract \( x \), and then add 4 to find the result. Understanding how these functions operate is foundational in algebra, as they are a vehicle through which we can model and solve problems ranging from simple calculations to complex real-world scenarios.