A function is given. Determine the average rate of change of the function between the given values of the variable.

f(x) = 6 − x2; x = 2, x = 2 + h

To determine the average rate of change of a function between two values of the variable, we use the formula:

Average rate of change = (f(x₂) - f(x₁)) / (x₂ - x₁)

In this case, we have:

f(x) = 6 − x^2
x₁ = 2
x₂ = 2 + h

We need to find f(x₂) and substitute the values into the formula.

Step 1: Substitute x = 2 + h into the function to get f(x₂):
f(x₂) = 6 − (2 + h)^2

Step 2: Simplify the expression inside the parentheses:
f(x₂) = 6 − (4 + 4h + h^2)

Step 3: Expand the squared term:
f(x₂) = 6 − 4 - 4h - h^2

Step 4: Simplify the expression further:
f(x₂) = 2 - 4h - h^2

Now, we can substitute the values into the formula:

Average rate of change = (f(x₂) - f(x₁)) / (x₂ - x₁)
= (2 - 4h - h^2 - f(x₁)) / (x₂ - x₁)
= (2 - 4h - h^2 - (6 - x₁^2)) / (x₂ - x₁)
= (2 - 4h - h^2 - (6 - 4)) / (x₂ - x₁)
= (6 - 4h - h^2 - 2) / (x₂ - x₁)
= (4 - 4h - h^2) / h

Thus, the average rate of change of the function between x = 2 and x = 2 + h is (4 - 4h - h^2) / h.

To determine the average rate of change of a function, we need to find the difference between the function values at two different points and divide it by the difference between the corresponding values of the variable. In this case, we have the function f(x) = 6 - x^2 and two values of x: x = 2 and x = 2 + h, where h is some small change in x.

Step 1: Find the function value at x = 2:
f(2) = 6 - (2)^2 = 6 - 4 = 2

Step 2: Find the function value at x = 2 + h:
f(2 + h) = 6 - (2 + h)^2 = 6 - (4 + 4h + h^2) = 6 - 4 - 4h - h^2 = 2 - 4h - h^2

Step 3: Find the average rate of change:
The average rate of change of the function between x = 2 and x = 2 + h is given by:
(Average Rate of Change) = (Change in Function Values) / (Change in Variable Values)

Change in Function Values = f(2 + h) - f(2)
= (2 - 4h - h^2) - 2
= - 4h - h^2

Change in Variable Values = (2 + h) - 2
= h

Average Rate of Change = (- 4h - h^2) / h

Step 4: Simplify the expression:
Average Rate of Change = - 4 - h

Therefore, the average rate of change of the function f(x) = 6 - x^2 between x = 2 and x = 2 + h is given by -4 - h.