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Polynomial functions are a type of mathematical function that take a form like this: \(f(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0\). They consist of terms with non-negative integer exponents. Each term includes a coefficient and a variable raised to a power.
For example, in the function \(f(x) = x^2 - x + 4\), there are three terms. The first term is \(x^2\), making it a quadratic polynomial because the highest power is 2. The coefficients here are 1, -1, and 4.
The degree of the polynomial is determined by the highest exponent, which tells us about the behavior of the function, especially as \(x\) gets very large or very small.
Polynomial functions are versatile and appear in many areas of mathematics because they can closely approximate other functions and model a wide range of real-world phenomena.

Function evaluation means finding the output of a function for a particular input. Let's use the function \(g(x) = 3x - 5\) as an example.
To evaluate \(g(x)\) at a specific value, simply replace the variable \(x\) with the given value.

• If asked to evaluate \(g(-1)\), you substitute -1 into the equation: \(g(-1) = 3(-1) - 5\).
• Calculating this gives \(g(-1) = -3 - 5 = -8\).

This process helps us understand the function's behavior with specific inputs, both visually (in graphing) and practically (in solving problems).
Evaluating functions is a fundamental skill because it allows us to predict outcomes and make informed decisions based on function models.

The substitution method is a strategy used in mathematics to evaluate complex functions by replacing variables.
It is especially useful in function compositions where a function is nested within another function. Consider a scenario where you have two functions, like \(f(x)\) and \(g(x)\), and need to find \(f(g(x))\).

• First, calculate \(g(x)\) at the desired point. For example, calculate \(g(-1)\) gives \(-8\).
• Next, substitute this result into \(f(x)\). Here, substitute \(-8\) into \(f\) to get \(f(-8)\).

Continuing with our example, \(f(-8) = (-8)^2 - (-8) + 4 = 76\).
This method is powerful for simplifying problems and calculating outcomes of layered functions, providing a clear path to follow when solving function composition problems.