In this image, the function $f(x) = x$ is defined, and the interval is given as $[0, 2]$. The expression for the average rate of change of the function, $\frac{\Delta y}{\Delta x}$, seems to be incomplete, so we can compute it.

The average rate of change of a function over an interval $[a, b]$ is given by: $\frac{\Delta y}{\Delta x} = \frac{f(b) - f(a)}{b - a}$

For the function $f(x) = x$ over the interval $[0, 2]$:

• $f(0) = 0$
• $f(2) = 2$

Thus, the average rate of change is: $\frac{\Delta y}{\Delta x} = \frac{f(2) - f(0)}{2 - 0} = \frac{2 - 0}{2 - 0} = \frac{2}{2} = 1$

So, the average rate of change of the function $f(x) = x$ over the interval $[0, 2]$ is 1.

Would you like further details or clarification on this? Here are some related questions you might find useful:

1. What is the geometric interpretation of the average rate of change for a linear function?
2. How does the average rate of change differ for non-linear functions?
3. Can the average rate of change be negative? What does it represent in such cases?
4. How does the rate of change vary over different intervals for a quadratic function?
5. How do we interpret the instantaneous rate of change compared to the average rate of change?

Tip: The average rate of change of a linear function is always constant, which is equal to the slope of the line.