Given the function:

$f(x) = 9x - 7$

with $x = 2$ and $x = 3$, let's proceed to solve the problems.

(a) Net Change Between the Given Values of the Variable

The net change in the function is the difference in function values at $x = 3$ and $x = 2$. This is calculated as:

$\text{Net Change} = f(3) - f(2)$

Now, we can calculate $f(3)$ and $f(2)$:

$f(3) = 9(3) - 7 = 27 - 7 = 20$

$f(2) = 9(2) - 7 = 18 - 7 = 11$

Thus, the net change is:

$\text{Net Change} = 20 - 11 = 9$

(b) Average Rate of Change Between the Given Values of the Variable

The average rate of change is given by the formula:

$\text{Average Rate of Change} = \frac{f(3) - f(2)}{3 - 2}$

We already know the net change is 9, and the difference in the $x$-values is $3 - 2 = 1$. So:

$\text{Average Rate of Change} = \frac{9}{1} = 9$

Final Answers:

• (a) Net Change: 9
• (b) Average Rate of Change: 9

Would you like further details or clarification on these calculations?

Here are 5 related questions:

1. What does the net change represent in terms of the graph of the function?
2. How would the net change change if the function were quadratic?
3. How is the average rate of change different from the instantaneous rate of change?
4. How can you use the average rate of change to approximate the slope of the tangent line to the function?
5. What happens to the average rate of change if the function's coefficients are altered?

Tip: The average rate of change can be interpreted as the slope of the secant line between two points on the function’s graph.