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Substitution is a fundamental technique in evaluating functions. It means replacing the variable with a given number. For example, in the function \( f(x) = -3x + 1 \), to find \( f(3) \), you substitute 3 for \( x \). So, you will perform the following steps: Use the equation \( f(x) = -3x + 1 \).
Replace \( x \) with 3, thus getting \( f(3) = -3(3) + 1 \).
Simplify to get \( f(3) = -9 + 1 = -8 \).
This gives us the value of \( f \) at \( x = 3 \).

The same principle applies to \( g(x) \). If \( g(x) = x^2 + 2 \), then \( g(3) = (3)^2 + 2 \).
This results in \( g(3) = 9 + 2 = 11 \).
Substitution is particularly useful when dealing with polynomial functions, as it allows you to simplify and solve them easily. Remember, just plug in the value and simplify.

Division of functions involves finding the quotient of two functions evaluated at a certain point. After finding the values of \( f \) and \( g \) at a specific point, proceed to divide them. For our exercise, after finding \( f(3) = -8 \) and \( g(3) = 11 \), divide \( f(3) \) by \( g(3) \). The division is straightforward: \[ \frac{f(3)}{g(3)} = \frac{-8}{11} \].
This quotient tells you the ratio of the two function values. It's a common operation in higher-level algebra and calculus. Understanding how to divide functions is important for solving more complex mathematical problems.

Remember, when performing division, ensure the denominator is not zero to avoid undefined results.

Polynomial functions are expressions involving variables raised to whole number powers. They are characterized by their degrees, which are the highest power of the variable. In our example, \( g(x) = x^2 + 2 \) is a polynomial function of degree 2, since the highest power is \( x^2 \).

Polynomials are versatile and appear frequently in mathematics. When evaluating polynomial functions like \( g(x) \), follow a similar substitution method:
Given \( g(x) = x^2 + 2 \), substitute the chosen value: \( g(3) = (3)^2 + 2 \), and simplify: \( g(3) = 9 + 2 = 11 \).
Polynomials can be simple or complex, and understanding their evaluation is crucial for progressing in math.

Other common examples include linear functions \( f(x) \) like in our exercise, or more complex higher-degree polynomials. Always substitute and simplify step-by-step to avoid mistakes.