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Linear functions are one of the simplest types of functions you will encounter, and they are characterized by creating a straight line when graphed on a coordinate plane. These functions have a form similar to \( f(x) = mx + b \). In this equation, the letter \( m \) represents the slope of the line, and \( b \) is known as the y-intercept.

• The slope \( m \) indicates how steep the line is and in which direction it goes. A positive slope means the line ascends from left to right, while a negative slope descends.
• The y-intercept \( b \) is the point where the line intersects the y-axis, meaning where it crosses the vertical line on a graph when \( x=0 \).

The linear function presented, \( f(x)=3x-2 \), has a slope of 3. This means that for every increase by 1 in \( x \), the function's output increases by 3. The y-intercept is -2, indicating the line crosses the y-axis at -2. This straightforward relationship between \( x \) and \( f(x) \) makes evaluating and graphing linear functions straightforward.

Function evaluation is the process of finding the output, or value, of a function given a specific input. When evaluating a function like \( f(x) = 3x - 2 \), you simply substitute the input value into the function and calculate the result.
Here’s how you evaluate a function step-by-step:

• Identify the function and its specific input values. For example, in our problem, the function is \( f(x) = 3x - 2 \) and we need to evaluate at \( x = 2 \) and \( x = 3 \).
• Substitute each input into the function. For \( x = 2 \), replace \( x \) with 2: \( f(2) = 3(2) - 2 \).
• Simplify the expression to find the output. In this case: \( f(2) = 6 - 2 = 4 \).
• Repeat the process for \( x = 3 \): \( f(3) = 3(3) - 2 = 9 - 2 = 7 \).

Through function evaluation, we have determined that for \( x = 2 \), the function outputs 4, and for \( x = 3 \), it outputs 7. This step is crucial for calculating the average rate of change.

Mathematical formulas are tools that help us solve problems by providing a structured method for computation. In this context, we used the average rate of change formula to find how the function changes between two points. This can be thought of as finding the slope of the line that connects these two points on the graph of a function.
The formula for the average rate of change is:

• \[ \frac{f(b) - f(a)}{b - a} \]

This represents the change in the function's value (\( f \)) over the change in the input (\( x \)).
To apply this formula:

• First, compute the function values \( f(a) \) and \( f(b) \) for the given inputs. In our exercise, they are 4 and 7 for \( x = 2 \) and \( x = 3 \), respectively.
• Next, calculate the difference in function values: \( f(b) - f(a) = 7 - 4 \).
• Then, find the change in the inputs: \( b - a = 3 - 2 \).
• Finally, divide the function's value change by the input change: \( \frac{3}{1} = 3 \).

The average rate of change formula revealed that as \( x \) increases from 2 to 3, the function \( f(x) = 3x - 2 \) increases by 3 units, consistent with the slope of the line.