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For the linear function f(x)=0.5x + 6, find the domain, range, and intercepts. Write the domain and range in interval notation. Then find the minimum and maximum values of f(x) on the given interval.

f(x) = 0.5x + 6, [-2,6]

User Klmdb
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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.

User Fcm
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