Final answer:
Option 3: 4 evaluating f(1) for either equation gives us f(1) = 2(1) + 5 = 7 or f(1) = -2(1) + 10 = 8, reaffirming the value 1 exists in the domain of f(x)f(x). ""
Explanation:
The domain of a function refers to the set of all possible input values (x) for which the function is defined. In this case, the function f(x) is defined as f(x) = 2x + 5 for the first segment of the domain, and f(x) = -2x + 10 for the second segment. To find the values included in the domain of f(x)f(x), it's crucial to consider where both segments of the function are defined simultaneously. The only constraint to ensure both segments are defined is that the function should not involve division by zero or square roots of negative numbers, which isn't an issue in this scenario.
To find the common values in the domain, set both segments equal to each other and solve for x: 2x + 5 = -2x + 10. Solving this equation yields x = 1, indicating that x = 1 is the point where both segments of the function are defined. Plugging x = 1 into either segment confirms that f(1) exists for both equations, hence the value 1 lies within the domain.
Finally, evaluating f(1) for either equation gives us f(1) = 2(1) + 5 = 7 or f(1) = -2(1) + 10 = 8, reaffirming the value 1 exists in the domain of f(x)f(x). ""