Properties of the Cube Root Function

Oct 13, 2025 by talynt.org 146 views

Understanding the Behavior of the Function $g(x)=\sqrt[3]{x+6}$

In this article, we will delve into the properties of the given function $g(x)=\sqrt[3]{x+6}$ and determine the interval on which the function is negative. To begin with, let's understand the nature of the cube root function and its behavior.

The Cube Root Function

The cube root function, denoted by $\sqrt[3]{x}$ , is a function that takes a real number $x$ as input and returns the cube root of that number. The cube root of a number is a value that, when multiplied by itself twice (or cubed), gives the original number. For example, the cube root of 27 is 3, because $3^3 = 27$ .

Properties of the Cube Root Function

The cube root function has several important properties that are essential to understanding its behavior. One of the key properties is that the cube root function is an odd function, meaning that $\sqrt[3]{-x} = -\sqrt[3]{x}$ for all real numbers $x$ . This property implies that the cube root function is symmetric about the origin, and its graph is a reflection of the graph of the cube root function about the y-axis.

The Function $g(x)=\sqrt[3]{x+6}$

Now, let's consider the specific function $g(x)=\sqrt[3]{x+6}$ . This function is a transformation of the cube root function, where the input $x$ is shifted by 6 units to the left. In other words, the function $g(x)$ is equivalent to the cube root function with an input of $x+6$ .

Finding the Interval Where the Function is Negative

To determine the interval where the function $g(x)$ is negative, we need to find the values of $x$ for which $g(x) < 0$ . Since the cube root function is an odd function, we know that $\sqrt[3]{x+6} < 0$ if and only if $x+6 < 0$ . Therefore, we need to find the values of $x$ for which $x+6 < 0$ .

Solving the Inequality

To solve the inequality $x+6 < 0$ , we can subtract 6 from both sides, which gives us $x < -6$ . This means that the function $g(x)$ is negative for all values of $x$ that are less than -6.

Conclusion

In conclusion, the function $g(x)=\sqrt[3]{x+6}$ is negative on the interval $(-\infty, -6)$ . This interval represents all values of $x$ that are less than -6, and it is the only interval on which the function is negative.

Answer

The correct answer is A. $(-\infty, -6)$ .

Additional Insights

It's worth noting that the function $g(x)$ is not defined for $x=-6$ , since the cube root of zero is not defined. However, the function is defined for all values of $x$ that are greater than -6, and it is positive on this interval.

Graphical Representation

The graph of the function $g(x)$ is a cube root curve that is shifted 6 units to the left. The graph is a reflection of the cube root curve about the y-axis, and it is negative on the interval $(-\infty, -6)$ .

Real-World Applications

The function $g(x)=\sqrt[3]{x+6}$ has several real-world applications, including modeling population growth and decay, and analyzing the behavior of physical systems. In these applications, the function is often used to represent the rate of change of a quantity over time.

Conclusion

In this article, we have explored the properties of the function $g(x)=\sqrt[3]{x+6}$ and determined the interval on which the function is negative. We have also discussed the real-world applications of the function and provided a graphical representation of its behavior.