What is the range of the function f(x) = 3 - 2√(x - 1)?
Show the answer and explanation
Correct answer
A. (-∞, 3]
Principle or equation
The square root function √(x-1) has range [0, ∞). Multiplying by -2 gives (-∞, 0]. Adding 3 gives (-∞, 3].
Why this answer is correct
Domain: x ≥ 1. As x increases from 1 to ∞, √(x-1) increases from 0 to ∞, so -2√(x-1) decreases from 0 to -∞. Thus f(x) = 3 - 2√(x-1) takes all values from 3 down to -∞. Therefore range is (-∞, 3].
Example
For g(x) = 1 - √(x), range is (-∞, 1].
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