What is the range of $f(x)=4 \sqrt{2 x-1}+3$?
A. $[3, \infty)$
B. $(-\infty, 4)$
C. $(-\infty, 4]$
D. $(4, \infty)$
Answer (2)
Determine the domain of the function: xg e 2 1 .
Find the minimum value of the function by substituting x = 2 1 into f ( x ) , resulting in f ( 2 1 ) = 3 .
Observe that as x increases, f ( x ) also increases without bound.
Conclude that the range of f ( x ) is [ 3 , ∞ ) .
Explanation
Understanding the Problem The function is f ( x ) = 4 2 x − 1 + 3 . We want to find the range of this function.
Finding the Domain The domain of the function is determined by the square root. The expression inside the square root must be non-negative, so we have 2 x − 1 ≥ 0 . Solving for x , we get 2 x ≥ 1 , so x ≥ 2 1 .
Finding the Minimum Value Now let's find the range. Since x ≥ 2 1 , the smallest value of x is 2 1 . When x = 2 1 , we have f ( 2 1 ) = 4 2 ( 2 1 ) − 1 + 3 = 4 1 − 1 + 3 = 4 0 + 3 = 4 ( 0 ) + 3 = 3 .
Analyzing the Function's Behavior As x increases from 2 1 , the value of 2 x − 1 also increases, so 2 x − 1 increases. Therefore, 4 2 x − 1 increases, and f ( x ) = 4 2 x − 1 + 3 also increases. As x approaches infinity, f ( x ) also approaches infinity.
Determining the Range Since the minimum value of f ( x ) is 3, and f ( x ) increases without bound as x increases, the range of f ( x ) is [ 3 , ∞ ) .
Examples
Understanding the range of functions is crucial in various fields. For instance, in physics, if f ( x ) represents the velocity of an object where x is time, knowing the range tells us the possible velocity values the object can attain. If the range is [ 3 , ∞ ) , it means the object's velocity will always be greater than or equal to 3 units, which could be meters per second, kilometers per hour, etc. This helps in predicting and analyzing the object's motion.
The range of the function f ( x ) = 4 2 x − 1 + 3 is [ 3 , ∞ ) because the minimum value occurs at x = 2 1 and increases indefinitely as x increases. Therefore, the answer is option A.
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