The average rate of change of a function g(x) between two points a and b is calculated using the formula:
Average Rate of Change = (g(b) – g(a)) / (b – a)
In this case, we are given:
- a = 4
- b = 7
- Average Rate of Change = 5/6
According to the formula, we can set it up with our values:
5/6 = (g(7) – g(4)) / (7 – 4)
Since 7 – 4 = 3, we can simplify this further:
5/6 = (g(7) – g(4)) / 3
Now we can multiply both sides by 3 to eliminate the denominator:
3 * (5/6) = g(7) – g(4)
This simplifies to:
15/6 = g(7) – g(4)
Reducing 15/6 gives us:
5/2 = g(7) – g(4)
From this, we can conclude that:
- g(7) – g(4) = 2.5, which indicates that the difference in the outputs of the function g(x) at x=7 and x=4 is 2.5.
Thus, the statement that must be true is that the value of g(7) is 2.5 units greater than the value of g(4). However, keep in mind that while this conclusion holds for the given average rate of change, we cannot determine the actual values of g(4) or g(7) without additional information about the function g(x) itself.