- The function \(f(x) = -2\sqrt[3]{x+7}\) is defined for all real numbers.
- As \(x \to \infty\), the inner term \(x+7 \to \infty\), which makes \(\sqrt[3]{x+7} \to \infty\). Multiplying by \(-2\) results in \(f(x) \to -\infty\).
- As \(x \to -\infty\), the inner term \(x+7 \to -\infty\), which makes \(\sqrt[3]{x+7} \to -\infty\). Multiplying by \(-2\) results in \(f(x) \to \infty\).
- The correct description is: As \(x\) approaches positive infinity, \(f(x)\) approaches negative infinity. \(\boxed{\text{As } x \text{ approaches positive infinity, } f(x) \text{ approaches negative infinity.}}\)
Explanation
- Domain Analysis
The function provided is \(f(x) = -2\sqrt[3]{x+7}\). This is a transformation of the parent cube root function \(y = \sqrt[3]{x}\). The domain of any cube root function \(\sqrt[3]{u}\) is all real numbers, because any real number has a real cube root. Therefore, the function \(f(x)\) is defined for all \(x \in \mathbb{R}\).
- Analyzing the Inner Expression
To determine the end behavior, we look at what happens to \(f(x)\) as \(x\) becomes very large (positive infinity) and very small (negative infinity).
The inner expression is \(u = x+7\).
- As \(x \to \infty\), \(x+7 \to \infty\).
- As \(x \to -\infty\), \(x+7 \to -\infty\).
- Determining End Behavior
The parent function \(\sqrt[3]{u}\) has the property that:
- As \(u \to \infty\), \(\sqrt[3]{u} \to \infty\).
- As \(u \to -\infty\), \(\sqrt[3]{u} \to -\infty\).
Now, we apply the scalar multiplier \(-2\).
- As \(x \to \infty\), \(u \to \infty\), so \(\sqrt[3]{x+7} \to \infty\). Multiplying by \(-2\) gives \(-2(\infty) = -\infty\).
- As \(x \to -\infty\), \(u \to -\infty\), so \(\sqrt[3]{x+7} \to -\infty\). Multiplying by \(-2\) gives \(-2(-\infty) = \infty\).
- Conclusion
Based on our analysis:
- As \(x \to \infty\), \(f(x) \to -\infty\).
- As \(x \to -\infty\), \(f(x) \to \infty\).
Comparing these results with the given options, the statement "As \(x\) approaches positive infinity, \(f(x)\) approaches negative infinity" is correct.
Examples
Consider the trajectory of a specialized projectile whose path is modeled by a cube root function. Understanding end behavior allows engineers to predict the long-term trend of the projectile's height or distance, ensuring it doesn't cross restricted zones or exceed safety boundaries as it continues its motion towards extreme values of the independent variable, such as distance or time. Mathematical models help visualize these trends before a real-world test, minimizing risks and optimizing performance.