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Understanding how to substitute values into functions is crucial. A function can be thought of as a machine that takes an input (often represented as x) and produces an output by performing specific operations on the input.
To substitute a value into a function, follow these steps:

• Identify the function and the given input value.
• Replace the variable in the function with the given input value.
• Perform the operations as specified by the function to find the output.

For example, with the function \( f(x) = -2x + 3 \), to find \( f(2) \), you replace x with 2:
\[ f(2) = -2(2) + 3 = -4 + 3 = -1 \]
This means when x is 2, the output of the function f(x) is -1. Similarly, substituting 2 into \( g(x) = x^{2} - 5 \), we get:
\[ g(2) = 2^{2} - 5 = 4 - 5 = -1 \]
After you have substituted into both functions, you can then combine the results as needed.

Basic algebra helps us simplify and solve equations and functions. Here are a few key concepts:

• Substitution: Replace variables with numbers.
• Arithmetic Operations: Addition, subtraction, multiplication, and division.
• Simplification: Combine like terms and simplify expressions.

For instance, in the problem provided, once we substitute and evaluate, we need to handle the arithmetic correctly:
\[ f(2) - g(2) = -1 - (-1) \]
This step involves recognizing that subtracting a negative number is the same as adding its positive counterpart, hence:
\[ f(2) - g(2) = -1 + 1 = 0 \]
This demonstrates how basic algebra principles, like simplifying expressions and understanding the operations, are applied.

Polynomials are expressions formed by variables and coefficients using operations of addition, subtraction, multiplication, and non-negative integer exponents. In algebra, polynomials are fundamental components.
Let's break down the polynomial functions in the exercise:

• \(f(x) = -2x + 3\) is a linear polynomial, as the variable x is to the power of 1.
• \(g(x) = x^{2} - 5\) is a quadratic polynomial, where the highest degree of the variable x is 2.

Polynomials can vary greatly in form and complexity, but understanding the basics of evaluating polynomial functions by substitution, as shown in the exercise, is essential. Here, both functions are simplified after substituting x with 2, yielding manageable results:
\[ f(2) = -2(2) + 3 = -1 \]
\[ g(2) = (2)^2 - 5 = -1 \]
Recognizing the type of polynomial and then accurately performing the substitution and simplification apply across many algebra problems dealing with polynomials.