Final answer:
The domain of the function is all real numbers, the range is also all real numbers, the intercepts are -12 and 6, and the minimum and maximum values on the interval [-2,6] are 5 and 9 respectively.
Step-by-step explanation:
Domain:
The domain of a function represents the set of all possible input values, or x-values, for the function. In the case of the linear function f(x) = 0.5x + 6, there are no restrictions on the input values, so the domain is all real numbers (-∞, +∞).
Range:
The range of a function represents the set of all possible output values, or y-values, for the function. Since the function is a linear function with a positive slope, the range is also all real numbers (-∞, +∞).
Intercepts:
The x-intercept is the point at which the graph of the function intersects the x-axis. To find the x-intercept, set f(x) equal to zero and solve for x:
f(x) = 0.5x + 6
0 = 0.5x + 6
-6 = 0.5x
x = -12
So, the x-intercept is -12. The y-intercept is the point at which the graph of the function intersects the y-axis. The y-intercept occurs when x = 0:
f(0) = 0.5(0) + 6
f(0) = 6
So, the y-intercept is 6.
Minimum and Maximum Values:
To find the minimum and maximum values of the function on the given interval [-2,6], we evaluate the function at the endpoints of the interval and compare the values. Evaluating the function at x = -2:
f(-2) = 0.5(-2) + 6
f(-2) = 5
Evaluating the function at x = 6:
f(6) = 0.5(6) + 6
f(6) = 9
So, the minimum value of f(x) on the interval [-2,6] is 5 and the maximum value is 9.