Answer:
Step-by-step explanation:
This is a piecewise function. If we are finding the average rate of change over the interval 4 and 12 inclusive, that means that we are finding the slope of the whole function (both pieces) from 4 to 12 (domain values are from 4 to 12. That means x values). This is not a straight line, so the average value is merely a rough estimate of the slope between x values 4 and 12.
When x ≤ 4, we have to use the first piece of the function, 3x - 7 because the restriction on the domain is that x has to be less than 6 and 4 is less than 6. Evaluating at the domain of 4 in the first piece will give us an (x, y) coordinate to help find the slope.
f(4) = 3(4) - 7 and
f(4) = 12 - 7 so
f(4) = 5 and the coordinate is (4, 5).
When x ≤ 12 we have to use the second piece of the function, .75x + 10 because the domain restriction is that x has to be greater than or equal to 6 and 12 is greater than 6. Evaluating at a domain of 6 will give us the other coordinate we need to find the slope.
f(12) = .75(12) + 10 and
f(12) = 9 + 10 so
f(12) = 19 and the coordinate is (12, 19). Now for the slope (aka average rate of change). Using the slope formula:
[tex]m=\frac{y_2-y_1}{x_2-x_1}[/tex]
[tex]m=\frac{19-5}{12-4}=\frac{14}{8}=\frac{7}{4}[/tex]
There you go! Just remember the domain thing and you should be fine.
