Which Graph Represents The Function $f(x)=\sqrt[3]{x-2}$?A. B. C. D.

Introduction

Graphing functions is an essential skill in mathematics, and it's crucial to understand how to represent different types of functions on a coordinate plane. In this article, we'll focus on graphing the function $f(x)=\sqrt[3]{x-2}$, which involves a cube root. We'll explore the characteristics of this function, identify its key features, and determine which graph represents it.

Understanding the Function

The given function is $f(x)=\sqrt[3]{x-2}$. To begin, let's break down the components of this function:

• The cube root function, denoted by $\sqrt[3]{x}$, is a function that takes a number as input and returns its cube root.
• The expression $x-2$ inside the cube root function indicates that the input value is shifted 2 units to the right.

Key Features of the Function

To graph the function $f(x)=\sqrt[3]{x-2}$, we need to identify its key features:

• Domain: The domain of a function is the set of all possible input values. For the cube root function, the domain is all real numbers, but since we have $x-2$ inside the cube root, the domain is restricted to $x \geq 2$.
• Range: The range of a function is the set of all possible output values. Since the cube root function is increasing, the range is all real numbers.
• Asymptote: The asymptote of a function is a horizontal line that the function approaches as the input value increases without bound. In this case, the asymptote is the x-axis, $y=0$.
• Intercepts: The x-intercept is the point where the function crosses the x-axis, which occurs when $y=0$. To find the x-intercept, we set $f(x)=0$ and solve for $x$: $\sqrt[3]{x-2}=0 \Rightarrow x-2=0 \Rightarrow x=2$. The y-intercept is the point where the function crosses the y-axis, which occurs when $x=0$. To find the y-intercept, we substitute $x=0$ into the function: $f(0)=\sqrt[3]{0-2}=-\sqrt[3]{2}$.

Graphing the Function

Now that we've identified the key features of the function, let's graph it. We'll use the following steps:

1. Plot the asymptote: The asymptote is the x-axis, $y=0$.
2. Plot the x-intercept: The x-intercept is the point $(2,0)$.
3. Plot the y-intercept: The y-intercept is the point $(0,-\sqrt[3]{2})$.
4. Plot additional points: To get a better idea of the graph, we can plot additional points by substituting different values of $x$ into the function.

Which Graph Represents the Function?

Now that we've graphed the function, let's compare it to the given options:

A. Graph 1: This graph has a vertical asymptote at $x=2$ and a horizontal asymptote at $y=0$. It also has an x-intercept at $(2,0)$ and a y-intercept at $(0,-\sqrt[3]{2})$. However, the graph is not increasing, which contradicts the fact that the cube root function is increasing.

B. Graph 2: This graph has a horizontal asymptote at $y=0$ and an x-intercept at $(2,0)$. However, it does not have a y-intercept at $(0,-\sqrt[3]{2})$.

C. Graph 3: This graph has a horizontal asymptote at $y=0$ and an x-intercept at $(2,0)$. It also has a y-intercept at $(0,-\sqrt[3]{2})$. However, the graph is not increasing, which contradicts the fact that the cube root function is increasing.

D. Graph 4: This graph has a horizontal asymptote at $y=0$ and an x-intercept at $(2,0)$. It also has a y-intercept at $(0,-\sqrt[3]{2})$. Moreover, the graph is increasing, which is consistent with the fact that the cube root function is increasing.

Based on the analysis, we can conclude that Graph 4 represents the function $f(x)=\sqrt[3]{x-2}$.

Conclusion

Graphing functions is an essential skill in mathematics, and it's crucial to understand how to represent different types of functions on a coordinate plane. In this article, we've explored the characteristics of the function $f(x)=\sqrt[3]{x-2}$, identified its key features, and determined which graph represents it. By following the steps outlined in this article, you can graph any function and identify its key features.

Introduction

In our previous article, we explored the characteristics of the function $f(x)=\sqrt[3]{x-2}$, identified its key features, and determined which graph represents it. In this article, we'll answer some common questions related to graphing this function.

Q: What is the domain of the function $f(x)=\sqrt[3]{x-2}$?

A: The domain of the function $f(x)=\sqrt[3]{x-2}$ is all real numbers greater than or equal to 2, i.e., $x \geq 2$. This is because the cube root function is defined for all real numbers, but the expression $x-2$ inside the cube root function restricts the domain to $x \geq 2$.

Q: What is the range of the function $f(x)=\sqrt[3]{x-2}$?

A: The range of the function $f(x)=\sqrt[3]{x-2}$ is all real numbers. This is because the cube root function is increasing, and the expression $x-2$ inside the cube root function does not restrict the range.

Q: What is the asymptote of the function $f(x)=\sqrt[3]{x-2}$?

A: The asymptote of the function $f(x)=\sqrt[3]{x-2}$ is the x-axis, $y=0$. This is because the cube root function approaches the x-axis as the input value increases without bound.

Q: What are the x-intercept and y-intercept of the function $f(x)=\sqrt[3]{x-2}$?

A: The x-intercept of the function $f(x)=\sqrt[3]{x-2}$ is the point $(2,0)$, and the y-intercept is the point $(0,-\sqrt[3]{2})$. This is because the function crosses the x-axis at $x=2$ and the y-axis at $y=-\sqrt[3]{2}$.

Q: How do I graph the function $f(x)=\sqrt[3]{x-2}$?

A: To graph the function $f(x)=\sqrt[3]{x-2}$, follow these steps:

1. Plot the asymptote, which is the x-axis, $y=0$.
2. Plot the x-intercept, which is the point $(2,0)$.
3. Plot the y-intercept, which is the point $(0,-\sqrt[3]{2})$.
4. Plot additional points by substituting different values of $x$ into the function.

Q: Which graph represents the function $f(x)=\sqrt[3]{x-2}$?

A: Based on our previous analysis, the graph that represents the function $f(x)=\sqrt[3]{x-2}$ is Graph 4.

Conclusion

Graphing functions is an essential skill in mathematics, and it's crucial to understand how to represent different types of functions on a coordinate plane. In this article, we've answered some common questions related to graphing the function $f(x)=\sqrt[3]{x-2}$. By following the steps outlined in this article, you can graph any function and identify its key features.

Frequently Asked Questions

• Q: What is the cube root function? A: The cube root function is a function that takes a number as input and returns its cube root.
• Q: What is the domain of the cube root function? A: The domain of the cube root function is all real numbers.
• Q: What is the range of the cube root function? A: The range of the cube root function is all real numbers.
• Q: What is the asymptote of the cube root function? A: The asymptote of the cube root function is the x-axis, $y=0$.

Additional Resources

• Graphing Functions: A comprehensive guide to graphing functions, including the cube root function.
• Mathematics Tutorials: A collection of tutorials on various mathematical topics, including graphing functions.
• Mathematics Resources: A list of resources for learning mathematics, including textbooks, online courses, and practice problems.