Compare The Given Graphs Of F ( X ) = 2 X F(x)=2 \sqrt{x} F ( X ) = 2 X And G ( X ) = 2 X 3 G(x)=2 \sqrt[3]{x} G ( X ) = 2 3 X And Determine Which Of The Following Is True:A. The Graphs Have Different Domains.B. They Are Both Decreasing On Their Domains.C. When $x \ \textgreater \

Mar 12, 2025 by ADMIN 287 views

Comparing the Graphs of $f(x)=2 \sqrt{x}$ and $g(x)=2 \sqrt[3]{x}$

In mathematics, functions are used to describe the relationship between variables. Graphs are a visual representation of these functions, providing valuable insights into their behavior. In this article, we will compare the graphs of two functions, $f(x)=2 \sqrt{x}$ and $g(x)=2 \sqrt[3]{x}$ , and determine which of the given statements is true.

Understanding the Functions

The function $f(x)=2 \sqrt{x}$ is a square root function, where the input $x$ is under the square root sign. This function is also known as the radical function. The function $g(x)=2 \sqrt[3]{x}$ is a cube root function, where the input $x$ is under the cube root sign.

Domain and Range

The domain of a function is the set of all possible input values for which the function is defined. The range of a function is the set of all possible output values.

For the function $f(x)=2 \sqrt{x}$ , the domain is all non-negative real numbers, i.e., $x \geq 0$ . This is because the square root of a negative number is not defined in the real number system. The range of this function is all non-negative real numbers, i.e., $y \geq 0$ .

For the function $g(x)=2 \sqrt[3]{x}$ , the domain is all real numbers, i.e., $x \in \mathbb{R}$ . This is because the cube root of any real number is defined. The range of this function is also all real numbers, i.e., $y \in \mathbb{R}$ .

Graphs of the Functions

The graph of a function is a visual representation of the function, showing the relationship between the input and output values.

The graph of $f(x)=2 \sqrt{x}$ is a curve that opens upwards, with the x-axis as the asymptote. The graph starts at the origin (0,0) and increases as x increases.

The graph of $g(x)=2 \sqrt[3]{x}$ is a curve that also opens upwards, but with a different shape than the graph of $f(x)$ . The graph of $g(x)$ has a more gradual increase than the graph of $f(x)$ .

Comparing the Graphs

Now that we have a good understanding of the functions and their graphs, let's compare the two graphs.

The graph of $f(x)=2 \sqrt{x}$ has a more rapid increase than the graph of $g(x)=2 \sqrt[3]{x}$ . This is because the square root function grows faster than the cube root function.

Determining the Truth of the Statements

Now that we have compared the graphs of the two functions, let's determine which of the given statements is true.

A. The graphs have different domains.

The graph of $f(x)=2 \sqrt{x}$ has a domain of all non-negative real numbers, i.e., $x \geq 0$ . The graph of $g(x)=2 \sqrt[3]{x}$ has a domain of all real numbers, i.e., $x \in \mathbb{R}$ . Therefore, the graphs have different domains.

B. They are both decreasing on their domains.

The graph of $f(x)=2 \sqrt{x}$ is an increasing function, meaning that as x increases, y also increases. The graph of $g(x)=2 \sqrt[3]{x}$ is also an increasing function. Therefore, they are not both decreasing on their domains.

C. When $x \ \textgreater \ 0$ , the graph of $f(x)$ is above the graph of $g(x)$ .

When $x \ \textgreater \ 0$ , the graph of $f(x)=2 \sqrt{x}$ is above the graph of $g(x)=2 \sqrt[3]{x}$ . This is because the square root function grows faster than the cube root function.

Conclusion

In conclusion, the graphs of $f(x)=2 \sqrt{x}$ and $g(x)=2 \sqrt[3]{x}$ have different domains, and the graph of $f(x)$ is above the graph of $g(x)$ when $x \ \textgreater \ 0$ . Therefore, statement A is true, and statement C is also true.

References

[1] "Functions" by Khan Academy
[2] "Graphs of Functions" by Math Open Reference
[3] "Square Root and Cube Root Functions" by Purplemath
Q&A: Comparing the Graphs of $f(x)=2 \sqrt{x}$ and $g(x)=2 \sqrt[3]{x}$

In our previous article, we compared the graphs of two functions, $f(x)=2 \sqrt{x}$ and $g(x)=2 \sqrt[3]{x}$ , and determined which of the given statements is true. In this article, we will answer some frequently asked questions about the comparison of these two functions.

Q: What is the main difference between the graphs of $f(x)$ and $g(x)$ ?

A: The main difference between the graphs of $f(x)$ and $g(x)$ is the rate at which they increase. The graph of $f(x)=2 \sqrt{x}$ increases more rapidly than the graph of $g(x)=2 \sqrt[3]{x}$ .

Q: Why do the graphs of $f(x)$ and $g(x)$ have different domains?

A: The graph of $f(x)=2 \sqrt{x}$ has a domain of all non-negative real numbers, i.e., $x \geq 0$ , because the square root of a negative number is not defined in the real number system. The graph of $g(x)=2 \sqrt[3]{x}$ has a domain of all real numbers, i.e., $x \in \mathbb{R}$ , because the cube root of any real number is defined.

Q: Are the graphs of $f(x)$ and $g(x)$ both increasing on their domains?

A: Yes, both graphs are increasing on their domains. The graph of $f(x)=2 \sqrt{x}$ is an increasing function, meaning that as x increases, y also increases. The graph of $g(x)=2 \sqrt[3]{x}$ is also an increasing function.

Q: When $x \ \textgreater \ 0$ , is the graph of $f(x)$ above the graph of $g(x)$ ?

A: Yes, when $x \ \textgreater \ 0$ , the graph of $f(x)=2 \sqrt{x}$ is above the graph of $g(x)=2 \sqrt[3]{x}$ . This is because the square root function grows faster than the cube root function.

Q: What is the significance of the comparison between the graphs of $f(x)$ and $g(x)$ ?

A: The comparison between the graphs of $f(x)$ and $g(x)$ is significant because it highlights the difference in the rate of increase between the square root and cube root functions. This difference is important in various mathematical and real-world applications.

Q: Can the comparison between the graphs of $f(x)$ and $g(x)$ be extended to other functions?

A: Yes, the comparison between the graphs of $f(x)$ and $g(x)$ can be extended to other functions. For example, we can compare the graphs of $f(x)=2 \sqrt{x}$ and $g(x)=2 \sqrt[4]{x}$ to see how the rate of increase changes.

Conclusion

In conclusion, the comparison between the graphs of $f(x)=2 \sqrt{x}$ and $g(x)=2 \sqrt[3]{x}$ is an important one, highlighting the difference in the rate of increase between the square root and cube root functions. We hope that this Q&A article has provided valuable insights into this comparison.

References

[1] "Functions" by Khan Academy
[2] "Graphs of Functions" by Math Open Reference
[3] "Square Root and Cube Root Functions" by Purplemath