Select the correct answer.
Which function has a range of $[-6, \infty)$ ?
$f(x)=\sqrt{x-4}+6$
$f(x)=\sqrt{x+4}-6$
$f(x)=\sqrt{x+6}-4$
$f(x)=\sqrt{x-6}+4$
Answer (2)
The range of a function f ( x ) = x + a + b is [ b , ∞ ) .
We are looking for a function with a range of [ − 6 , ∞ ) .
The function f ( x ) = x + 4 − 6 has a range of [ − 6 , ∞ ) .
Therefore, 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 general form of the functions is f ( x ) = x + a + b , where a and b are constants. The square root function x has a range of [ 0 , ∞ ) . Therefore, x + a also has a range of [ 0 , ∞ ) . Adding a constant b to x + a shifts the range to [ b , ∞ ) . We need to find the function where b = − 6 .
Analyzing the first function The range of f ( x ) = x − 4 + 6 is [ 6 , ∞ ) .
Analyzing the second function The range of f ( x ) = x + 4 − 6 is [ − 6 , ∞ ) .
Analyzing the third function The range of f ( x ) = x + 6 − 4 is [ − 4 , ∞ ) .
Analyzing the fourth function The range of f ( x ) = x − 6 + 4 is [ 4 , ∞ ) .
Conclusion The function f ( x ) = x + 4 − 6 has the range [ − 6 , ∞ ) . Therefore, the correct answer is f ( x ) = x + 4 − 6 .
Examples
Understanding function ranges is crucial in many real-world applications. For instance, when designing a suspension bridge, engineers need to ensure that the cables can handle a certain range of weights. Similarly, in economics, understanding the range of a cost function can help businesses determine the minimum and maximum costs they might incur. This problem demonstrates how transformations affect the range of a function, a concept widely used in modeling and optimization problems.
The function that has a range of [ − 6 , ∞ ) is f ( x ) = x + 4 − 6 . This is because the square root outputs start from 0, and subtracting 6 results in a minimum output of -6. Thus, the correct answer is f ( x ) = x + 4 − 6 .
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