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In mathematics, a function describes a relationship between inputs and outputs. For any given input, the function provides exactly one output. To evaluate a function, you replace the variable with a given number to calculate its result. For example, we have a function \( f(x) = -x^{3} + 6x^{2} + x \).

To find specific outputs, substitute values like \( x_1 = 1 \) or \( x_2 = 6 \) into the function. This process is known as "function evaluation." For \( x_1 = 1 \), replace \( x \) with 1:

\[ f(1) = -(1)^{3} + 6(1)^{2} + 1 \]

Calculate it to get \( f(1) = 6 \). Do the same for \( x_2 = 6 \) to get \( f(6) = 216 \).

These function evaluations give you the necessary outputs to work with in other calculations, like finding the average rate of change.

Polynomial functions are mathematical expressions involving a sum of powers of variables multiplied by coefficients. They can represent various phenomena in math and science. An example of a polynomial function is \( f(x) = -x^{3} + 6x^{2} + x \).

Each term in a polynomial has a coefficient (a number) and a power (an exponent) of \( x \). In our example:

• \(-x^{3}\) is a term with a coefficient of \(-1\).
• \(6x^{2}\) has a coefficient of 6.
• \(x\) is the linear term with a coefficient of 1.

Polynomials are easy to evaluate at a given point by replacing \( x \) with the value you want.

Understanding these functions is essential for analyzing how variables affect the output. They help in solving real-world problems involving rates, trends, and changes.

The rate of change formula lets you analyze how a function's output changes between two points. It's particularly useful in determining trends, like speed or growth. We use the average rate of change formula when evaluating functions between specified input values.

For the formula, find \( f(x_1) \) and \( f(x_2) \), then use:

\[ \text{Average Rate of Change} = \frac{f(x_2) - f(x_1)}{x_2 - x_1} \]

This formula essentially measures the "slope" between the two points on the function's graph.

In our exercise, evaluate the function at \( x_1 = 1 \) and \( x_2 = 6 \), then calculate:

• \( f(1) = 6 \)
• \( f(6) = 216 \)

Plug these values into the formula:

\[ \frac{216 - 6}{6 - 1} = 42 \]

This outcome shows that the function's output increases at an average rate of 42 units per unit interval over the specified range. Understanding how to calculate this helps in various fields such as physics, engineering, and economics to assess rates and changes.