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Substitution is a fundamental technique in mathematics that involves replacing a variable in an expression with a specific value or another expression. This process is essential for evaluating functions at given points. When you have a function like \( g(x) = x^{2} - 10x - 3 \), substitution allows you to compute specific values by replacing \( x \) with the numbers or expressions provided.Here's how it works:

• Start with your function: \( g(x) = x^{2} - 10x - 3 \).
• For the example where you are asked to find \( g(-1) \), substitute \(-1\) for every \( x \) in the function. Your new expression becomes \( (-1)^2 - 10(-1) - 3 \).
• Similarly, if asked to evaluate at \( g(x+2) \), replace all \( x \)s with \( (x+2) \), creating \( (x+2)^{2} - 10(x+2) - 3 \).
• If needed to find \( g(-x) \), substitute \(-x\) in place of \( x \), leading to \( (-x)^{2} - 10(-x) - 3 \).

Substitution simplifies the complex tasks of algebra by turning expressions with variables into plain numbers or smaller algebraic ones, aiding in further simplification.

Once you substitute values into a function, the next step is simplifying the expression. Simplifying makes the expression easier to use and interpret. Let's look at this through our function example \( g(x) = x^{2} - 10x - 3 \).After substitution, you end up with expressions like \( (-1)^2 - 10(-1) - 3 \) for \( g(-1) \). Here’s a step-by-step way to simplify:

• Solve the exponents: Calculate \((-1)^2\) which results in 1.
• Handle the multiplication or division next: Calculate \(-10 \times -1\), which results in 10.
• Proceed with addition and subtraction: Combine numbers, \(1 + 10 - 3\), resulting in 8.

You apply the same technique when simplifying \( g(x+2) \) and \( g(-x) \). It's crucial to combine like terms, which may involve adding or subtracting similar variables or constant terms. Simplifying expressions reduces their complexity, making them easier to work with and understand.

Polynomial functions are a core component of algebra. They comprise sums of terms, each consisting of a variable raised to a non-negative integer power, including a coefficient. An example of a polynomial function is \( g(x) = x^{2} - 10x - 3 \).Characteristics of polynomial functions include:

• They are smooth and continuous graphs.
• The degree of the polynomial indicates its highest power, which determines the curve's shape.
• They can be categorized into linear, quadratic, cubic, etc., based on their degree.

When you evaluate a polynomial function like \( g(x) \) at different values or expressions, you're looking into the specific points or transformations that it undergoes. For example, finding \( g(x+2) \) results in a transformation of the original function based on the substitution. Understanding polynomial functions allows one to predict and describe the behavior of a wide range of mathematical and real-world problems.