Select the correct answer.
Which statement is true about this radical function?
[tex]$f(x)=-\sqrt{x+6}$[/tex]
A. As [tex]$x$[/tex] approaches positive infinity, [tex]$f(x)$[/tex] approaches positive infinity.
B. As [tex]$x$[/tex] approaches negative infinity, [tex]$f(x)$[/tex] approaches positive infinity.
C. As [tex]$x$[/tex] approaches positive infinity, [tex]$f(x)$[/tex] approaches negative infinity.
D. As [tex]$x$[/tex] approaches negative infinity, [tex]$f(x)$[/tex] approaches negative infinity.
Answer (2)
Determine the domain of the function: x ≥ − 6 .
Analyze the behavior as x approaches positive infinity: x + 6 approaches positive infinity.
Consider the negative sign: f ( x ) = − x + 6 approaches negative infinity.
The correct statement is: As x approaches positive infinity, f ( x ) approaches negative infinity. C
Explanation
Understanding the Problem The problem asks us to determine the behavior of the function f ( x ) = − x + 6 as x approaches positive or negative infinity. First, we need to consider the domain of the function.
Determining the Domain The expression inside the square root must be non-negative, so we have x + 6 ≥ 0 , which means x ≥ − 6 . Therefore, the domain of the function is [ − 6 , ∞ ) . This tells us that x cannot approach negative infinity, since the function is not defined for x < − 6 .
Analyzing the Behavior as x Approaches Positive Infinity Now, let's analyze the behavior of the function as x approaches positive infinity. As x gets larger and larger, x + 6 also gets larger and larger, approaching positive infinity. The square root of a number that approaches positive infinity also approaches positive infinity. So, x + 6 approaches positive infinity as x approaches positive infinity.
Considering the Negative Sign However, our function is f ( x ) = − x + 6 , which means we have a negative sign in front of the square root. Therefore, as x approaches positive infinity, f ( x ) approaches negative infinity.
Conclusion Based on our analysis, the correct statement is: As x approaches positive infinity, f ( x ) approaches negative infinity.
Examples
Understanding the behavior of radical functions is crucial in various fields, such as physics and engineering. For instance, when analyzing the motion of an object under the influence of gravity, the distance the object falls is often modeled by a square root function. Knowing how the function behaves as time increases (approaches positive infinity) helps engineers predict the object's position and velocity. Similarly, in electrical engineering, the current flowing through a circuit can sometimes be modeled by a radical function, and understanding its behavior helps in designing efficient and safe circuits.
The function f ( x ) = − x + 6 is defined for x ≥ − 6 . As x approaches positive infinity, f ( x ) approaches negative infinity. Therefore, the correct answer is option C .
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