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Understanding function evaluation is crucial for working with mathematical functions in algebra. To evaluate a function means to calculate the output for a given input. When we have a function, such as f(x) = 14x - 3, and we want to find f(3), we are simply substituting the input value (3 in this case) into the function.

Here's how it works: Replace every instance of x in the function with the value 3. It looks like this: f(3) = 14 * 3 - 3. We then simplify the expression to get the result, which is the evaluated value of the function at x = 3.

Quick Tips for Function Evaluation:

• Always replace the variable with the given number, enclosed in parentheses.
• Perform the arithmetic operations in the correct order: multiplication or division before addition or subtraction.
• Double-check your calculation to avoid simple errors.

Polynomial functions are algebraic expressions that consist of variables and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables. An example is g(x) = 8x^2, which is also the function used in our exercise.

This particular function is a degree 2 polynomial, often called a quadratic function. The highest exponent of the variable x determines the degree. Evaluating polynomial functions, such as finding g(3), follows the same basic principles as other function evaluations. Substituting 3 for x gives us g(3) = 8 * (3^2), which simplifies to 72.

Keep in mind that the shape of the graph of polynomial functions varies with the degree and the coefficients. Quadratics, for example, always form a parabola.

When we talk about the difference of functions in algebra, we're referring to subtracting one function from another. The notation (g - f)(x) represents the function created when f(x) is subtracted from g(x). To find the value of (g - f)(3), as in our exercise, you first find the individual values g(3) and f(3), and then subtract the latter from the former.

The operation works at any point where both functions are defined, and it's important to keep in mind that the order matters—(g - f)(x) is not the same as (f - g)(x) because subtraction is not commutative. In our example, this resulted in (g - f)(3) = 72 - 39 = 33.

Understanding how functions combine through operations like subtraction enables you to tackle more complex problems and understand the behavior of polynomial functions in relation to each other.

Algebraic expressions are combinations of constants, variables, and algebraic operations like addition, subtraction, multiplication, and division. They are fundamental in expressing relationships and solving equations in algebra. Both functions we've discussed, f(x) and g(x), are examples of algebraic expressions.

One key skill in working with algebraic expressions is simplification, which involves combining like terms and performing operations to write the expression in its most reduced form. While simplifying, always be mindful of the order of operations and special rules, such as the distributive property.

In our problem, evaluating the function and finding the difference required that we treat each part as an individual algebraic expression. By simplifying accurately, we arrived at the correct answers for our functions’ evaluations and their difference.