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Substitution is one of the key methods in algebra. It helps us compute the value of a function at a specific point. When you have a function and you need to find its value for a particular variable, you simply replace the variable with the given number. This process is called substitution.
For example, given the function \( g(x) = -x^2 + 4x + 1 \) and asked to find \( g(-1) \), we substitute \( x \) with -1. This results in: \[ g(-1) = -(-1)^2 + 4(-1) + 1 \]
Here, ensure you correctly replace every instance of \( x \) in the function with -1. This step is crucial as it affects every part of the expression. Substitution transforms abstract expressions into specific numbers, making them easier to work with. It's a simple yet powerful tool in algebra.

Function evaluation involves calculating the value of a function for specific values of its variables. It's like finding how the function behaves at a particular point.
For instance, given a function \( f(x) = -3x + 4 \), if you need to evaluate \( f(2) \), you substitute \( x \) with 2, and then calculate the result: \[ f(2) = -3(2) + 4 = -6 + 4 = -2 \]
In our original problem, we were asked to evaluate \( g(-1) \). After substituting -1 into \( g(x) \), we simplify the expression step by step. This means breaking it down into smaller parts and solving in a sequence. First, calculate the power, then multiply and add as required by the function's operations. Each evaluation provides a specific snapshot of the function's output.

Algebraic expressions are combinations of variables, numbers, and operators (like addition and multiplication). They form the backbone of algebra and are used to represent real-world situations mathematically.
For example, the function \( g(x) = -x^2 + 4x + 1 \) is an algebraic expression. It consists of terms like \( -x^2 \), \( 4x \), and 1, each contributing to the overall expression.
When working with algebraic expressions, simplification is key. In the problem we solved, after substituting, we simplified the expression: \[ g(-1) = -(-1)^2 + 4(-1) + 1 \] This meant calculating powers, products, and sums step by step: \[ -1 + (-4) + 1 = -4 \]
Simplifying expressions makes them easier to understand and work with. It reduces complex problems into manageable parts, often revealing the underlying structure.