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What are the domain and range of ( f(x) = |x - 3| + 6 )?

a) Domain: x | x is all real numbers, Range: y | y > 6
b) Domain: x | x > 3, Range: y | y > 6
c) Domain: x | x is all real numbers, Range: y | 2 < y < 6
d) Domain: x | x < 3, Range: y | 2 < y < 6

1 Answer

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Final answer:

The domain of the function f(x) = |x - 3| + 6 is all real numbers, and the range is y such that y is greater than or equal to 6.

Step-by-step explanation:

The function in question is f(x) = |x - 3| + 6. To determine the domain and range of this function, we need to consider the properties of the absolute value function and any transformations applied to it. The absolute value function, |x|, is defined for all real numbers, so the expression inside the absolute value, x - 3, is also defined for all real numbers. This implies that the domain of f(x) is all real numbers. As for the range, since the absolute value is always non-negative, |x - 3| is always greater than or equal to 0. Adding 6 to it shifts the graph vertically upwards by 6 units, making the smallest value of f(x) equal to 6. Hence, f(x) is greater than or equal to 6 for all x, and there is no upper limit. Therefore, the range of f(x) is y | y ≥ 6.

Based on this analysis, the correct answer to the question is:

Domain: x | x is all real numbers, Range: y | y ≥ 6.

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