Identify The Function With The Following Characteristics:- Vertical Asymptote At $x=4$- Horizontal Asymptote At $y=2$- The $x$-intercept And $y$-intercept Are At The Origin.A. $f(x)=\frac{2x}{x-4}$B.

In mathematics, functions are used to describe the relationship between variables. Identifying functions with specific characteristics is an essential skill in mathematics, as it helps us understand the behavior of functions and make predictions about their behavior. In this article, we will discuss how to identify a function with a vertical asymptote at $x=4$, a horizontal asymptote at $y=2$, and the $x$-intercept and $y$-intercept at the origin.

Understanding the Characteristics of a Function

Before we dive into identifying the function, let's understand the characteristics mentioned in the problem.

• Vertical Asymptote: A vertical asymptote is a vertical line that the function approaches but never touches. In this case, the vertical asymptote is at $x=4$, which means that the function approaches infinity as $x$ approaches 4 from the left or right.
• Horizontal Asymptote: A horizontal asymptote is a horizontal line that the function approaches as $x$ approaches infinity. In this case, the horizontal asymptote is at $y=2$, which means that the function approaches 2 as $x$ approaches infinity.
• x-intercept and y-intercept at the Origin: The $x$-intercept is the point where the function crosses the $x$-axis, and the $y$-intercept is the point where the function crosses the $y$-axis. In this case, both intercepts are at the origin, which means that the function passes through the point (0,0).

Analyzing the Options

Now that we understand the characteristics of the function, let's analyze the options given.

Option A: $f(x)=\frac{2x}{x-4}$

This function has a vertical asymptote at $x=4$, as the denominator becomes zero when $x=4$. However, the horizontal asymptote is not at $y=2$, as the function approaches infinity as $x$ approaches infinity. Therefore, this function does not match the given characteristics.

Option B: $f(x)=\frac{2}{x}+2$

This function has a horizontal asymptote at $y=2$, as the function approaches 2 as $x$ approaches infinity. However, the vertical asymptote is not at $x=4$, as the function approaches infinity as $x$ approaches 0. Therefore, this function does not match the given characteristics.

Option C: $f(x)=\frac{2x}{x-4}+2$

This function has a vertical asymptote at $x=4$, as the denominator becomes zero when $x=4$. The horizontal asymptote is at $y=2$, as the function approaches 2 as $x$ approaches infinity. The $x$-intercept and $y$-intercept are at the origin, as the function passes through the point (0,0). Therefore, this function matches the given characteristics.

Conclusion

In conclusion, the function that matches the given characteristics is $f(x)=\frac{2x}{x-4}+2$. This function has a vertical asymptote at $x=4$, a horizontal asymptote at $y=2$, and the $x$-intercept and $y$-intercept at the origin.

Key Takeaways

• A vertical asymptote is a vertical line that the function approaches but never touches.
• A horizontal asymptote is a horizontal line that the function approaches as $x$ approaches infinity.
• The $x$-intercept is the point where the function crosses the $x$-axis, and the $y$-intercept is the point where the function crosses the $y$-axis.
• To identify a function with specific characteristics, we need to analyze the function's behavior and compare it with the given characteristics.

Further Reading

For further reading on functions and their characteristics, we recommend the following resources:

• Khan Academy: Functions and Graphs
• MIT OpenCourseWare: Calculus
• Wolfram MathWorld: Functions

In our previous article, we discussed how to identify a function with a vertical asymptote at $x=4$, a horizontal asymptote at $y=2$, and the $x$-intercept and $y$-intercept at the origin. In this article, we will answer some frequently asked questions about identifying functions with specific characteristics.

Q: What is a vertical asymptote?

A: A vertical asymptote is a vertical line that the function approaches but never touches. In other words, it is a line that the function gets arbitrarily close to but never crosses.

Q: What is a horizontal asymptote?

A: A horizontal asymptote is a horizontal line that the function approaches as $x$ approaches infinity. In other words, it is a line that the function gets arbitrarily close to as $x$ gets larger and larger.

Q: What is the difference between a vertical asymptote and a hole in a graph?

A: A vertical asymptote is a vertical line that the function approaches but never touches, whereas a hole in a graph is a point where the function is not defined but approaches a specific value. In other words, a vertical asymptote is a line that the function gets arbitrarily close to but never crosses, whereas a hole is a point where the function is not defined but approaches a specific value.

Q: How do I determine the vertical asymptote of a function?

A: To determine the vertical asymptote of a function, you need to find the values of $x$ that make the denominator of the function equal to zero. These values of $x$ are the vertical asymptotes of the function.

Q: How do I determine the horizontal asymptote of a function?

A: To determine the horizontal asymptote of a function, you need to compare the degrees of the numerator and denominator of the function. If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is $y=0$. If the degree of the numerator is equal to the degree of the denominator, the horizontal asymptote is the ratio of the leading coefficients. If the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote.

Q: What is the significance of the x-intercept and y-intercept in a function?

A: The $x$-intercept is the point where the function crosses the $x$-axis, and the $y$-intercept is the point where the function crosses the $y$-axis. These intercepts are important because they help us understand the behavior of the function and make predictions about its behavior.

Q: How do I determine the x-intercept and y-intercept of a function?

A: To determine the $x$-intercept and $y$-intercept of a function, you need to set the function equal to zero and solve for $x$ and $y$, respectively.

Q: What is the difference between a function with a vertical asymptote and a function with a hole in the graph?

A: A function with a vertical asymptote is a function that approaches infinity as $x$ approaches a specific value, whereas a function with a hole in the graph is a function that is not defined at a specific point but approaches a specific value.

Q: How do I determine if a function has a vertical asymptote or a hole in the graph?

A: To determine if a function has a vertical asymptote or a hole in the graph, you need to analyze the function's behavior and compare it with the given characteristics.

Conclusion

In conclusion, identifying functions with specific characteristics is an essential skill in mathematics. By understanding the characteristics of functions and analyzing the options, we can identify the function that matches the given characteristics. This skill is essential in mathematics, as it helps us understand the behavior of functions and make predictions about their behavior.

Key Takeaways

• A vertical asymptote is a vertical line that the function approaches but never touches.
• A horizontal asymptote is a horizontal line that the function approaches as $x$ approaches infinity.
• The $x$-intercept is the point where the function crosses the $x$-axis, and the $y$-intercept is the point where the function crosses the $y$-axis.
• To identify a function with specific characteristics, we need to analyze the function's behavior and compare it with the given characteristics.

Further Reading

For further reading on functions and their characteristics, we recommend the following resources:

• Khan Academy: Functions and Graphs
• MIT OpenCourseWare: Calculus
• Wolfram MathWorld: Functions

By understanding the characteristics of functions and analyzing the options, we can identify the function that matches the given characteristics. This skill is essential in mathematics, as it helps us understand the behavior of functions and make predictions about their behavior.