Select the correct answer.
Which function has a range of $[-6, \infty)$ ?
A. $f(x)=\sqrt{x-4}+6$
B. $f(x)=\sqrt{x+4}-6$
C. $f(x)=\sqrt{x+6}-4$
D. $f(x)=\sqrt{x-6}+4$
Answer (2)
The problem asks to identify the function with a range of [ − 6 , ∞ ) .
Analyze each function in the form f ( x ) = g ( x ) + c , where the range is [ c , ∞ ) .
Determine that f ( x ) = x + 4 − 6 has a range of [ − 6 , ∞ ) .
The correct answer is f ( x ) = x + 4 − 6 .
Explanation
Understanding the Problem We are given four functions and asked to identify the one with a range of [ − 6 , ∞ ) . The functions are of the form f ( x ) = g ( x ) + c , where g ( x ) is a linear function and c is a constant. The square root function g ( x ) has a range of [ 0 , ∞ ) when g ( x ) ≥ 0 . Therefore, the range of f ( x ) = g ( x ) + c is [ c , ∞ ) . We need to find the function where c = − 6 .
Determining the Range of Each Function Let's examine each function and determine its range.
f ( x ) = x − 4 + 6 : Since the square root function has a range of [ 0 , ∞ ) , adding 6 shifts the range to [ 6 , ∞ ) .
f ( x ) = x + 4 − 6 : Similarly, the square root function has a range of [ 0 , ∞ ) , and subtracting 6 shifts the range to [ − 6 , ∞ ) .
f ( x ) = x + 6 − 4 : The range is [ − 4 , ∞ ) .
f ( x ) = x − 6 + 4 : The range is [ 4 , ∞ ) .
Identifying the Correct Function We are looking for the function with a range of [ − 6 , ∞ ) . From the above analysis, the function f ( x ) = x + 4 − 6 has the desired range.
Final Answer Therefore, the correct answer is f ( x ) = x + 4 − 6 .
Examples
Understanding function ranges is crucial in many real-world applications. For example, if you're designing a sensor that measures temperature, the range of the sensor's output function determines the possible temperature values it can accurately report. Similarly, in finance, the range of a stock's price function tells you the minimum and maximum possible values of the stock. Knowing how transformations affect the range of a function allows you to predict and control the possible outcomes in these scenarios.
The function with a range of [ − 6 , → infinity) is f ( x ) = x + 4 − 6 . Therefore, the correct choice is B . This function correctly matches the required range after evaluation.
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