Given G Of X Equals Cube Root Of The Quantity X Plus 6, On What Interval Is The Function Negative? (,

Given G Of X Equals Cube Root Of The Quantity X Plus 6, On What Interval Is The Function Negative?

Understanding the behavior of functions is fundamental in calculus and algebra, especially when analyzing their positivity and negativity over specific intervals. In this article, we focus on the function \( G(x) = \sqrt[3]{x + 6} \), exploring the intervals where this function takes negative values. We will delve into the properties of cube root functions, analyze their domain, and determine the exact interval where \( G(x) \) is less than zero.

Overview of the Cube Root Function

Definition of the Cube Root Function

The cube root function, denoted as \( \sqrt[3]{x} \), is a mathematical function that maps any real number \( x \) to its cube root. Unlike square roots, which are only defined for non-negative numbers, cube roots are defined for all real numbers because every real number has a unique real cube root.

Mathematically, the cube root function is expressed as:
\[
f(x) = \sqrt[3]{x}
\]
and it possesses the following properties:

• Domain: \( (-\infty, \infty) \)


• Range: \( (-\infty, \infty) \)


• Even/Odd: The cube root function is an odd function, satisfying \( \sqrt[3]{-x} = -\sqrt[3]{x} \).

Properties Relevant to Our Function \( G(x) \)

Our specific function is:
\[
G(x) = \sqrt[3]{x + 6}
\]
which is a shifted version of the basic cube root function. The properties that influence its negativity include:

• The domain remains all real numbers, as cube roots are defined everywhere.


• The function is continuous and strictly increasing across its entire domain.


• The sign of \( G(x) \) depends on the sign of its argument \( x + 6 \).

Analyzing the Function \( G(x) = \sqrt[3]{x + 6} \)

Domain of \( G(x) \)

Since the cube root function is defined for all real numbers, the domain of \( G(x) \) is: \[ \boxed{(-\infty, \infty)} \] regardless of the value of \( x \).

Range of \( G(x) \)

Similarly, because the cube root covers all real numbers, the range is: \[ \boxed{(-\infty, \infty)} \]

Sign of \( G(x) \)

The sign of \( G(x) \) depends on the sign of the expression inside the cube root: \[ x + 6 \]

• If \( x + 6 > 0 \), then \( G(x) > 0 \).
• If \( x + 6 = 0 \), then \( G(x) = 0 \).
• If \( x + 6 < 0 \), then \( G(x) < 0 \).

This leads to a straightforward criterion for the negativity of the function.

Determining When \( G(x) \) Is Negative

Setting Up the Inequality

To find the interval where \( G(x) \) is negative, we analyze: \[ G(x) < 0 \] which simplifies to: \[ \sqrt[3]{x + 6} < 0 \]

Solving the Inequality

Since the cube root function is strictly increasing, the inequality \( \sqrt[3]{x + 6} < 0 \) is equivalent to: \[ x + 6 < 0 \] because applying the cube root preserves the inequality's direction.

Solving this:
\[
x + 6 < 0 \implies x < -6
\]

Conclusion on the Interval

Therefore, the function \( G(x) \) is negative precisely when \( x < -6 \).

Summary of Results

| Condition | Inequality | Solution Interval |
|---|---|---|
| \( G(x) < 0 \) | \( \sqrt[3]{x + 6} < 0 \) | \( x < -6 \) |
| \( G(x) = 0 \) | \( \sqrt[3]{x + 6} = 0 \) | \( x = -6 \) |
| \( G(x) > 0 \) | \( \sqrt[3]{x + 6} > 0 \) | \( x > -6 \) |

Hence, the domain where the function \( G(x) \) is negative is \( (-\infty, -6) \).

Visual Representation of \( G(x) \)

Visualizing the function helps to understand its behavior intuitively.

Graph of \( G(x) = \sqrt[3]{x + 6} \)

• The graph is a shifted cube root curve, shifting the basic \( \sqrt[3]{x} \) to the left by 6 units.
• The point \( x = -6 \) corresponds to \( G(-6) = \sqrt[3]{0} = 0 \).
• For \( x < -6 \), the graph lies below the x-axis, indicating negative \( G(x) \).
• For \( x > -6 \), the graph lies above the x-axis, indicating positive \( G(x) \).

Applications and Significance

Understanding where \( G(x) \) is negative is essential in various contexts, including:

• Mathematical Analysis: Helps in solving inequalities involving cube root functions.


• Physics: When modeling phenomena where the cube root term represents some physical quantity that can be negative.


• Engineering: For signal processing or control systems where negative values indicate specific states.

Additional Insights and Related Concepts

Continuity and Differentiability

• \( G(x) \) is continuous and differentiable everywhere because cube root functions are smooth across their domain.
• The derivative:

\[ G'(x) = \frac{1}{3} (x + 6)^{-\frac{2}{3}} \] which exists for all \( x \neq -6 \), highlighting a potential point of non-differentiability at \( x = -6 \).

Behavior at \( x = -6 \)

• The function crosses zero at \( x = -6 \).
• The function is increasing across its entire domain, with no local maxima or minima.

Summary and Final Remarks

In conclusion, for the function \( G(x) = \sqrt[3]{x + 6} \), the interval on which the function is negative is:
\[
\boxed{(-\infty, -6)}
\]
This is derived directly from the properties of cube root functions and the inequality \( x + 6 < 0 \). Recognizing the relationship between the inside of the cube root and the sign of the function is key to solving such problems effectively.

Understanding the intervals of positivity and negativity of functions like \( G(x) \) is crucial in many mathematical applications, from solving inequalities to analyzing real-world data. The cube root's property of being defined for all real numbers simplifies the analysis compared to other roots, making it a valuable tool in various fields.