91Ó°ÊÓ

Functions play a key role in mathematics, providing a way to represent relationships between variables. In its simplest form, a function is like a machine where you input a number, and out pops another number based on the rule of the function. For instance, in the function given in the exercise, \( f(x) = 6x \), every input \( x \) is multiplied by 6 to give the output \( f(x) \). This tells you how one quantity depends on another.

Functions can come in different forms: linear, exponential, quadratic, and more. The function \( f(x) = 6x \) is a linear function, meaning as \( x \) increases, \( f(x) \) increases at a constant rate. This brings us to the concept of the average rate of change, which helps us understand how a quantity changes on average over an interval.

Linear equations describe straight lines when plotted on a graph. They are especially useful for describing relationships where change happens at a constant rate. The equation\( y = mx + b \) is a classic example of a linear equation, where \( m \) is the slope and \( b \) is the y-intercept. In our exercise, the function \( f(x) = 6x \) simplifies to a linear equation where \( m = 6 \) and \( b = 0 \).

Understanding linear equations is crucial because they simplify calculations and provide a clear picture of how two quantities relate over equal intervals. Each increment in \( x \) leads to a consistent increment in \( f(x) \), which in this case is 6 times the increment. This consistency is why linear equations are prevalent in numerous real-world applications, from economics to engineering.

The substitution method is a technique used to find unknown values in functions or equations. It involves replacing variables with given values to simplify problems. In the exercise, we are given two \( x \)-values: \( x_1 = 0 \) and \( x_2 = 4 \). By substituting these into the linear function \( f(x) = 6x \), we calculate the corresponding \( y \)-values.

Here's how this works:

• For \( x_1 = 0 \), substitute to get \( f(x_1) = 6 \times 0 = 0 \).
• For \( x_2 = 4 \), substitute to get \( f(x_2) = 6 \times 4 = 24 \).

This straightforward process allows us to transition from using abstract symbols to numerical solutions that can be easily manipulated to find further properties like the average rate of change.

Calculation steps are essential to solving mathematical problems as they provide a systematic approach to arriving at a solution. By detailing each stage of the calculation, we minimize errors and increase accuracy.

In this exercise, the calculation steps involve:
1. Identifying the variables and the function: Here, it's \( f(x) = 6x \) with \( x_1 = 0 \) and \( x_2 = 4 \).
2. Substituting into the function: Calculate \( f(x_1) = 0 \) and \( f(x_2) = 24 \).
3. Applying these values to the average rate of change formula \( m = (f(x_2) - f(x_1)) / (x_2 - x_1) \), leading to \( m = (24 - 0) / (4 - 0) \).
4. Final calculation to determine the average rate of change as \( m = 6 \).

These structured steps make complex problems more manageable, ensuring students can replicate the process for other similar tasks easily.