The Graph Of $f(x)=\sqrt[3]{x+8}$ Is Shown.Which Statement Is True?A. The Function Is Only Increasing When $x \geq -8$.B. The Function Is Only Increasing When $ X ≥ 0 X \geq 0 X ≥ 0 [/tex].C. The Function Is Always

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Introduction

When analyzing the graph of a function, it's essential to understand its behavior, including where it's increasing or decreasing. In this article, we'll delve into the graph of $f(x)=\sqrt[3]{x+8}$ and determine which statement is true regarding its increasing behavior.

Understanding the Function

The given function is $f(x)=\sqrt[3]{x+8}$. This is a cubic root function, which means it has a unique characteristic: it's defined for all real numbers. The cubic root function is also known for its ability to handle negative numbers, unlike the square root function, which is only defined for non-negative numbers.

Analyzing the Graph

To analyze the graph of $f(x)=\sqrt[3]{x+8}$, we need to consider the behavior of the function as x varies. Since the function is a cubic root, it will always be non-negative, as the cube of any real number is non-negative.

Increasing Behavior

To determine where the function is increasing, we need to find the intervals where the function's derivative is positive. The derivative of $f(x)=\sqrt[3]{x+8}$ is $f'(x)=\frac{1}{3(x+8)^{\frac{2}{3}}}$.

Finding the Critical Points

To find the critical points, we need to set the derivative equal to zero and solve for x. In this case, we have:

13(x+8)23=0\frac{1}{3(x+8)^{\frac{2}{3}}} = 0

Since the denominator cannot be zero, we can multiply both sides by $3(x+8)^{\frac{2}{3}}$ to get:

1=01 = 0

This is a contradiction, which means there are no critical points. Therefore, the function is always increasing.

Conclusion

Based on the analysis, we can conclude that the function $f(x)=\sqrt[3]{x+8}$ is always increasing. This means that the correct statement is:

C. The function is always increasing.

Discussion

The graph of $f(x)=\sqrt[3]{x+8}$ is a simple yet interesting function. By analyzing its behavior, we can gain a deeper understanding of the properties of cubic root functions. In this case, we found that the function is always increasing, which is a unique characteristic of this type of function.

Final Thoughts

In conclusion, the graph of $f(x)=\sqrt[3]{x+8}$ is a fascinating function that exhibits unique behavior. By understanding its increasing behavior, we can gain a deeper appreciation for the properties of cubic root functions and their applications in mathematics and other fields.

References

Related Topics

Keywords

  • Cubic root function
  • Increasing behavior
  • Derivative
  • Critical points
  • Graph of a function
  • Mathematical analysis

Introduction

In our previous article, we analyzed the graph of $f(x)=\sqrt[3]{x+8}$ and determined that the function is always increasing. In this article, we'll answer some frequently asked questions about the graph of this function.

Q&A

Q1: What is the domain of the function $f(x)=\sqrt[3]{x+8}$?

A1: The domain of the function $f(x)=\sqrt[3]{x+8}$ is all real numbers, as the cubic root function is defined for all real numbers.

Q2: Is the function $f(x)=\sqrt[3]{x+8}$ always increasing?

A2: Yes, the function $f(x)=\sqrt[3]{x+8}$ is always increasing, as we determined in our previous article.

Q3: What is the range of the function $f(x)=\sqrt[3]{x+8}$?

A3: The range of the function $f(x)=\sqrt[3]{x+8}$ is all non-negative real numbers, as the cubic root function is always non-negative.

Q4: How do you find the derivative of the function $f(x)=\sqrt[3]{x+8}$?

A4: To find the derivative of the function $f(x)=\sqrt[3]{x+8}$, we use the power rule of differentiation, which states that if $f(x)=x^n$, then $f'(x)=nx^{n-1}$. In this case, we have $f(x)=(x+8)^{\frac{1}{3}}$, so $f'(x)=\frac{1}{3}(x+8)^{\frac{2}{3}}$.

Q5: What is the critical point of the function $f(x)=\sqrt[3]{x+8}$?

A5: There are no critical points for the function $f(x)=\sqrt[3]{x+8}$, as the derivative is always positive.

Q6: How do you graph the function $f(x)=\sqrt[3]{x+8}$?

A6: To graph the function $f(x)=\sqrt[3]{x+8}$, we can use a graphing calculator or software, such as Desmos or GeoGebra. We can also use the graphing features of a calculator or computer to visualize the graph.

Q7: What is the equation of the horizontal asymptote of the function $f(x)=\sqrt[3]{x+8}$?

A7: The equation of the horizontal asymptote of the function $f(x)=\sqrt[3]{x+8}$ is $y=0$, as the function approaches zero as x approaches negative infinity.

Q8: How do you find the equation of the vertical asymptote of the function $f(x)=\sqrt[3]{x+8}$?

A8: There are no vertical asymptotes for the function $f(x)=\sqrt[3]{x+8}$, as the function is defined for all real numbers.

Conclusion

In this article, we answered some frequently asked questions about the graph of $f(x)=\sqrt[3]{x+8}$. We hope this article has been helpful in understanding the behavior of this function.

References

Related Topics

Keywords

  • Cubic root function
  • Increasing behavior
  • Derivative
  • Critical points
  • Graph of a function
  • Mathematical analysis