- Determine the domain of \(f(x) = -\sqrt{x+6}\) to be \([-6, \infty)\), which implies that the function is undefined as \(x \to -\infty\).
- Analyze the limit as \(x \to \infty\), observing that \(\sqrt{x+6} \to \infty\), which implies \(-\sqrt{x+6} \to -\infty\).
- Conclude that the statement "As \(x\) approaches positive infinity, \(f(x)\) approaches negative infinity" is the only correct description of the function's end behavior.
- Final answer: \(\boxed{\text{As } x \text{ approaches positive infinity, } f(x) \text{ approaches negative infinity.}}\)
Explanation
- Analyzing the Domain and Defining the Scope
To analyze the radical function \(f(x) = -\sqrt{x+6}\), we must first determine its domain. The square root function \(\sqrt{u}\) is defined only for non-negative values of \(u\), so we require \(x + 6 \geq 0\). This inequality leads to \(x \geq -6\). Thus, the domain of \(f(x)\) is the interval \([-6, \infty)\). This immediately tells us that the function is not defined for values of \(x\) smaller than \(-6\), so discussing the limit as \(x\) approaches negative infinity is not applicable here.
- Evaluating the Limit as x Approaches Positive Infinity
We now evaluate the behavior of the function as \(x\) increases towards positive infinity. Since the domain is \([-6, \infty)\), we can investigate \(\lim_{x \to \infty} f(x) = \lim_{x \to \infty} -\sqrt{x+6}\). As \(x\) becomes very large, the expression \(x + 6\) also becomes very large. Consequently, the value of \(\sqrt{x+6}\) increases without bound, meaning \(\sqrt{x+6} \to \infty\) as \(x \to \infty\). Since the function is defined as the negative of this square root, it follows that \(-\sqrt{x+6} \to -\infty\) as \(x \to \infty\). Thus, as \(x\) approaches positive infinity, \(f(x)\) approaches negative infinity.
- Conclusion and Selection of the Correct Statement
Comparing our findings with the provided statements:
- "As \(x\) approaches positive infinity, \(f(x)\) approaches positive infinity." (Incorrect)
- "As \(x\) approaches negative infinity, \(f(x)\) approaches positive infinity." (Incorrect, as the function is not defined for \(x < -6\))
- "As \(x\) approaches positive infinity, \(f(x)\) approaches negative infinity." (Correct)
- "As \(x\) approaches negative infinity, \(f(x)\) approaches negative infinity." (Incorrect, as the function is not defined for \(x < -6\))
Therefore, the only true statement is that as \(x\) approaches positive infinity, \(f(x)\) approaches negative infinity.
Examples
Radical functions such as \(f(x) = -\sqrt{x+6}\) often represent real-world phenomena with boundaries, such as the relationship between the duration of a light source's decay and its intensity. For example, if you observe the remaining light intensity as a function of time \(x\), where \(x\) cannot be less than a certain threshold due to physical constraints, understanding the end behavior helps predict that the intensity will continuously drop to zero or below as time progresses. This mathematical modeling allows for safe and accurate predictions in various scientific experiments.