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In mathematics, function evaluation refers to the process of determining the output of a function for a particular input. Think of it like a machine where you put something in, and something else comes out based on a specific rule.
For example, if we have a function denoted by \( f(x) = 2x^2 \), it means for any value of \( x \), you need to square it first and then multiply the result by 2.

Here's a simple process you can follow when evaluating functions:

• Identify the function rule: this is the equation that defines the function, such as \( f(x) = 2x^2 \).
• Substitute the given value of \( x \) into the function. For instance, if \( x = 3 \), you would calculate \( f(3) = 2(3)^2 \).
• Perform the arithmetic to find the result. In our case, this leads to \( 2 \times 9 = 18 \).

Evaluating functions is a fundamental skill essential for understanding more complex mathematical concepts. It allows you to understand how different inputs affect the outcome of a function.
Through function evaluation, we make mathematical predictions and solve various practical problems.

Simplifying expressions is an important mathematical skill that involves reducing expressions to their simplest form. This often means combining like terms or factoring. When expressions are simpler, they are easier to work with and understand.
To simplify an expression, follow these steps:

• Identify like terms: These are the terms with the same variables and powers. For instance, in an expression \(3x + 2x \), both terms are alike because they involve \(x\).
• Combine like terms: Add or subtract the coefficients of the like terms. For example, \(3x + 2x = 5x\).
• Factor common terms if possible: Factoring involves rewriting an expression as a product of its factors. For instance, \(2x^2 - 18 \) can be factored as \(2(x^2 - 9)\).

In the exercise provided, simplifying involves not only combining terms but also canceling out terms in a fraction. To simplify \( \frac{2(x-3)(x+3)}{x-3} \), we cancel \( (x-3) \), leaving us with \( 2(x+3) \). Simplification like this is crucial for solving more complex equations efficiently.

Factoring polynomials is a process used to express a polynomial as the product of simpler polynomials. This is a key concept because it often helps simplify problems and solve equations.
When working on factoring, you'll frequently encounter the difference of squares, especially with quadratic polynomials like \(x^2 - a^2\).
Let's dive into the process of factoring:

• Identify and factor out the greatest common factor (GCF). For example, in \(2x^2 - 18\), the GCF is 2, so we factor it out: \(2(x^2 - 9)\).
• Recognize common patterns such as the difference of squares, \(x^2 - a^2 = (x-a)(x+a)\). This makes the factorization simpler.
• Once factored, these polynomials can be simplified in various expressions, making the overall problem easier to work with.

Correctly factoring a polynomial enables us to simplify expressions like \( \frac{2(x^2 - a^2)}{x-a} \) into \( 2(x+a) \), after canceling common terms. It is a powerful tool in the mathematical toolkit which lays the groundwork for solving higher degree equations.