Given $g(x) = \sqrt[3]{x-3}$, On What Interval Is The Function Positive?A. $(-\infty, -3)$B. $ ( − ∞ , 3 ) (-\infty, 3) ( − ∞ , 3 ) [/tex]C. $(-3, \infty)$D. $(3, \infty)$

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Introduction

In mathematics, the cubic root function is a fundamental concept that plays a crucial role in various mathematical operations. Given the function $g(x) = \sqrt[3]{x-3}$, we are tasked with determining the interval on which the function is positive. This requires a thorough understanding of the properties of the cubic root function and its behavior in different intervals.

Properties of the Cubic Root Function

The cubic root function, denoted by $\sqrt[3]{x}$, is defined as the inverse of the cube of a number. In other words, it returns the value that, when cubed, gives the original number. The cubic root function has several important properties that are essential in understanding its behavior:

  • The cubic root function is defined for all real numbers.
  • The cubic root function is an increasing function, meaning that as the input value increases, the output value also increases.
  • The cubic root function has a minimum value of 0, which occurs when the input value is 0.

Analyzing the Given Function

The given function $g(x) = \sqrt[3]{x-3}$ is a cubic root function with a horizontal shift of 3 units to the right. This means that the function is defined for all real numbers, but its behavior is affected by the shift.

To determine the interval on which the function is positive, we need to consider the behavior of the function in different intervals. Let's analyze the function in the following intervals:

  • (−∞,−3)( -\infty, -3 )

  • (−3,3)( -3, 3 )

  • (3,∞)( 3, \infty )

Interval $( -\infty, -3 )$

In this interval, the input value $x$ is less than -3. When we substitute $x = -3$ into the function, we get:

g(−3)=−3−33=−63g(-3) = \sqrt[3]{-3-3} = \sqrt[3]{-6}

Since the input value is negative, the output value is also negative. Therefore, the function is negative in the interval $( -\infty, -3 )$.

Interval $( -3, 3 )$

In this interval, the input value $x$ is between -3 and 3. When we substitute $x = 0$ into the function, we get:

g(0)=0−33=−33g(0) = \sqrt[3]{0-3} = \sqrt[3]{-3}

Since the input value is negative, the output value is also negative. Therefore, the function is negative in the interval $( -3, 3 )$.

Interval $( 3, \infty )$

In this interval, the input value $x$ is greater than 3. When we substitute $x = 4$ into the function, we get:

g(4)=4−33=13=1g(4) = \sqrt[3]{4-3} = \sqrt[3]{1} = 1

Since the input value is positive, the output value is also positive. Therefore, the function is positive in the interval $( 3, \infty )$.

Conclusion

Based on the analysis of the function in different intervals, we can conclude that the function $g(x) = \sqrt[3]{x-3}$ is positive in the interval $( 3, \infty )$. This means that the correct answer is:

  • D. $( 3, \infty )$

Therefore, the function $g(x) = \sqrt[3]{x-3}$ is positive on the interval $( 3, \infty )$.

Final Answer

Introduction

In our previous article, we explored the properties of the cubic root function and analyzed the given function $g(x) = \sqrt[3]{x-3}$ to determine the interval on which it is positive. In this article, we will address some frequently asked questions related to the cubic root function and its behavior in different intervals.

Q: What is the domain of the cubic root function?

A: The domain of the cubic root function is all real numbers. This means that the function is defined for any real input value.

Q: Is the cubic root function an increasing or decreasing function?

A: The cubic root function is an increasing function. This means that as the input value increases, the output value also increases.

Q: What is the minimum value of the cubic root function?

A: The minimum value of the cubic root function is 0, which occurs when the input value is 0.

Q: How does the horizontal shift affect the behavior of the cubic root function?

A: The horizontal shift affects the behavior of the cubic root function by changing the point at which the function is defined. In the given function $g(x) = \sqrt[3]{x-3}$, the horizontal shift is 3 units to the right. This means that the function is defined for all real numbers, but its behavior is affected by the shift.

Q: How do you determine the interval on which the function is positive?

A: To determine the interval on which the function is positive, you need to analyze the function in different intervals. You can do this by substituting different input values into the function and determining the sign of the output value.

Q: What is the interval on which the function $g(x) = \sqrt[3]{x-3}$ is positive?

A: Based on the analysis of the function in different intervals, we can conclude that the function $g(x) = \sqrt[3]{x-3}$ is positive in the interval $( 3, \infty )$.

Q: What is the correct answer?

A: The correct answer is D. $( 3, \infty )$.

Q: What are some common mistakes to avoid when working with the cubic root function?

A: Some common mistakes to avoid when working with the cubic root function include:

  • Not considering the domain of the function
  • Not analyzing the function in different intervals
  • Not determining the sign of the output value
  • Not considering the horizontal shift

Conclusion

In this article, we addressed some frequently asked questions related to the cubic root function and its behavior in different intervals. We also provided some tips on how to avoid common mistakes when working with the cubic root function. By understanding the properties of the cubic root function and analyzing the function in different intervals, you can determine the interval on which the function is positive.

Final Answer

The final answer is D. $( 3, \infty )$.

Additional Resources

For more information on the cubic root function and its behavior in different intervals, you can consult the following resources: