Let $f(x)=\frac{7}{x-9}$. Find The Domain, Vertical Asymptote(s), And Horizontal Asymptote.- Domain: $\square$ (Use Interval Notation)- Vertical Asymptote(s): $\square$ (Enter As A Comma-separated List If More Than One)-

Domain

To find the domain of the function $f(x)=\frac{7}{x-9}$, we need to determine the values of $x$ for which the function is defined. The function is defined as long as the denominator $x-9$ is not equal to zero. This is because division by zero is undefined.

We can find the values of $x$ for which the denominator is zero by setting $x-9=0$ and solving for $x$. This gives us:

$x-9=0$

$x=9$

Therefore, the function is undefined when $x=9$. This means that the domain of the function is all real numbers except $x=9$. In interval notation, this can be written as:

$(-\infty, 9) \cup (9, \infty)$

Vertical Asymptote(s)

A vertical asymptote is a vertical line that the graph of a function approaches but never touches. In the case of the function $f(x)=\frac{7}{x-9}$, the vertical asymptote is the line $x=9$. This is because the function is undefined when $x=9$, and the graph of the function approaches this line as $x$ approaches $9$ from either side.

Therefore, the vertical asymptote of the function is:

$x=9$

Horizontal Asymptote

A horizontal asymptote is a horizontal line that the graph of a function approaches as $x$ approaches infinity or negative infinity. In the case of the function $f(x)=\frac{7}{x-9}$, the horizontal asymptote is the line $y=0$. This is because as $x$ approaches infinity or negative infinity, the denominator $x-9$ approaches infinity, and the function approaches zero.

Therefore, the horizontal asymptote of the function is:

$y=0$

Conclusion

In conclusion, the domain of the function $f(x)=\frac{7}{x-9}$ is all real numbers except $x=9$, which can be written in interval notation as $(-\infty, 9) \cup (9, \infty)$. The vertical asymptote of the function is the line $x=9$, and the horizontal asymptote is the line $y=0$.

Step-by-Step Solution

Step 1: Find the domain of the function

To find the domain of the function, we need to determine the values of $x$ for which the function is defined. The function is defined as long as the denominator $x-9$ is not equal to zero. This is because division by zero is undefined.

We can find the values of $x$ for which the denominator is zero by setting $x-9=0$ and solving for $x$. This gives us:

$x-9=0$

$x=9$

Therefore, the function is undefined when $x=9$. This means that the domain of the function is all real numbers except $x=9$. In interval notation, this can be written as:

$(-\infty, 9) \cup (9, \infty)$

Step 2: Find the vertical asymptote(s) of the function

A vertical asymptote is a vertical line that the graph of a function approaches but never touches. In the case of the function $f(x)=\frac{7}{x-9}$, the vertical asymptote is the line $x=9$. This is because the function is undefined when $x=9$, and the graph of the function approaches this line as $x$ approaches $9$ from either side.

Therefore, the vertical asymptote of the function is:

$x=9$

Step 3: Find the horizontal asymptote of the function

A horizontal asymptote is a horizontal line that the graph of a function approaches as $x$ approaches infinity or negative infinity. In the case of the function $f(x)=\frac{7}{x-9}$, the horizontal asymptote is the line $y=0$. This is because as $x$ approaches infinity or negative infinity, the denominator $x-9$ approaches infinity, and the function approaches zero.

Therefore, the horizontal asymptote of the function is:

$y=0$

Final Answer

The final answer is:

Domain: $(-\infty, 9) \cup (9, \infty)$ Vertical Asymptote(s): $x=9$ Horizontal Asymptote: $y=0$

Q: What is the domain of the function $f(x)=\frac{7}{x-9}$?

A: The domain of the function is all real numbers except $x=9$. This can be written in interval notation as $(-\infty, 9) \cup (9, \infty)$.

Q: Why is the function undefined when $x=9$?

A: The function is undefined when $x=9$ because the denominator $x-9$ is equal to zero when $x=9$. Division by zero is undefined.

Q: What is the vertical asymptote of the function $f(x)=\frac{7}{x-9}$?

A: The vertical asymptote of the function is the line $x=9$. This is because the function is undefined when $x=9$, and the graph of the function approaches this line as $x$ approaches $9$ from either side.

Q: Why is there a vertical asymptote at $x=9$?

A: There is a vertical asymptote at $x=9$ because the function approaches infinity as $x$ approaches $9$ from either side. This is because the denominator $x-9$ approaches zero as $x$ approaches $9$, and the function approaches infinity as the denominator approaches zero.

Q: What is the horizontal asymptote of the function $f(x)=\frac{7}{x-9}$?

A: The horizontal asymptote of the function is the line $y=0$. This is because as $x$ approaches infinity or negative infinity, the denominator $x-9$ approaches infinity, and the function approaches zero.

Q: Why is there a horizontal asymptote at $y=0$?

A: There is a horizontal asymptote at $y=0$ because the function approaches zero as $x$ approaches infinity or negative infinity. This is because the denominator $x-9$ approaches infinity as $x$ approaches infinity or negative infinity, and the function approaches zero as the denominator approaches infinity.

Q: How do you find the domain, vertical asymptote(s), and horizontal asymptote of a rational function?

A: To find the domain, vertical asymptote(s), and horizontal asymptote of a rational function, you need to follow these steps:

1. Find the values of $x$ for which the denominator is zero. These values are the vertical asymptotes of the function.
2. Find the values of $x$ for which the function is undefined. These values are the vertical asymptotes of the function.
3. Determine the behavior of the function as $x$ approaches infinity or negative infinity. This will give you the horizontal asymptote of the function.

Q: What is the significance of the domain, vertical asymptote(s), and horizontal asymptote of a function?

A: The domain, vertical asymptote(s), and horizontal asymptote of a function are important because they help you understand the behavior of the function. The domain tells you where the function is defined, the vertical asymptote(s) tell you where the function approaches infinity, and the horizontal asymptote tells you where the function approaches a constant value.

Q: How do you use the domain, vertical asymptote(s), and horizontal asymptote of a function to graph the function?

A: To graph a function, you need to use the domain, vertical asymptote(s), and horizontal asymptote of the function. Here's how:

1. Plot the vertical asymptotes of the function on the graph.
2. Plot the horizontal asymptote of the function on the graph.
3. Use the domain of the function to determine the range of values of $x$ for which the function is defined.
4. Plot the function on the graph using the values of $x$ and $y$ that you have determined.

Q: What are some common mistakes to avoid when finding the domain, vertical asymptote(s), and horizontal asymptote of a function?

A: Some common mistakes to avoid when finding the domain, vertical asymptote(s), and horizontal asymptote of a function include:

• Not considering the values of $x$ for which the denominator is zero.
• Not considering the values of $x$ for which the function is undefined.
• Not determining the behavior of the function as $x$ approaches infinity or negative infinity.
• Not plotting the vertical and horizontal asymptotes on the graph.
• Not using the domain, vertical asymptote(s), and horizontal asymptote of the function to graph the function.

Q: How do you check your work when finding the domain, vertical asymptote(s), and horizontal asymptote of a function?

A: To check your work when finding the domain, vertical asymptote(s), and horizontal asymptote of a function, you need to follow these steps:

1. Check that you have found the correct values of $x$ for which the denominator is zero.
2. Check that you have found the correct values of $x$ for which the function is undefined.
3. Check that you have determined the correct behavior of the function as $x$ approaches infinity or negative infinity.
4. Check that you have plotted the vertical and horizontal asymptotes on the graph correctly.
5. Check that you have used the domain, vertical asymptote(s), and horizontal asymptote of the function to graph the function correctly.

Q: What are some real-world applications of the domain, vertical asymptote(s), and horizontal asymptote of a function?

A: Some real-world applications of the domain, vertical asymptote(s), and horizontal asymptote of a function include:

• Modeling population growth and decline.
• Modeling the spread of diseases.
• Modeling the behavior of electrical circuits.
• Modeling the behavior of mechanical systems.
• Modeling the behavior of financial systems.

Q: How do you use the domain, vertical asymptote(s), and horizontal asymptote of a function to solve real-world problems?

A: To use the domain, vertical asymptote(s), and horizontal asymptote of a function to solve real-world problems, you need to follow these steps:

1. Identify the problem and the function that models it.
2. Find the domain, vertical asymptote(s), and horizontal asymptote of the function.
3. Use the domain, vertical asymptote(s), and horizontal asymptote of the function to graph the function.
4. Use the graph of the function to solve the problem.

Q: What are some common challenges when finding the domain, vertical asymptote(s), and horizontal asymptote of a function?

A: Some common challenges when finding the domain, vertical asymptote(s), and horizontal asymptote of a function include:

• Finding the correct values of $x$ for which the denominator is zero.
• Finding the correct values of $x$ for which the function is undefined.
• Determining the correct behavior of the function as $x$ approaches infinity or negative infinity.
• Plotting the vertical and horizontal asymptotes on the graph correctly.
• Using the domain, vertical asymptote(s), and horizontal asymptote of the function to graph the function correctly.

Q: How do you overcome common challenges when finding the domain, vertical asymptote(s), and horizontal asymptote of a function?

A: To overcome common challenges when finding the domain, vertical asymptote(s), and horizontal asymptote of a function, you need to follow these steps:

1. Read the problem carefully and understand what is being asked.
2. Use algebraic techniques to find the values of $x$ for which the denominator is zero.
3. Use algebraic techniques to find the values of $x$ for which the function is undefined.
4. Use graphical techniques to determine the behavior of the function as $x$ approaches infinity or negative infinity.
5. Use graphical techniques to plot the vertical and horizontal asymptotes on the graph correctly.
6. Use the domain, vertical asymptote(s), and horizontal asymptote of the function to graph the function correctly.

Q: What are some advanced topics related to the domain, vertical asymptote(s), and horizontal asymptote of a function?

A: Some advanced topics related to the domain, vertical asymptote(s), and horizontal asymptote of a function include:

• Finding the domain, vertical asymptote(s), and horizontal asymptote of a function with multiple variables.
• Finding the domain, vertical asymptote(s), and horizontal asymptote of a function with complex numbers.
• Finding the domain, vertical asymptote(s), and horizontal asymptote of a function with matrices.
• Finding the domain, vertical asymptote(s), and horizontal asymptote of a function with differential equations.

Q: How do you apply the domain, vertical asymptote(s), and horizontal asymptote of a function to advanced topics?

A: To apply the domain, vertical asymptote(s), and horizontal asymptote of a function to advanced topics, you need to follow these steps:

1. Read the problem carefully and understand what is being asked.
2. Use algebraic techniques to find the values of $x$ for which the denominator is zero.
3. Use algebraic techniques to find the values of $x$ for which the function is undefined.
4. Use graphical techniques to determine the behavior of the function as $x$ approaches infinity or negative infinity.
5. Use graphical techniques to plot the vertical and horizontal asymptotes on the graph correctly.
6. Use the domain, vertical asymptote(s), and horizontal asymptote of the function to graph the function correctly.

Q: What are some real-world applications of advanced topics related to the domain, vertical asymptote(s), and horizontal asymptote of a function?

A: Some real-world applications of advanced topics related to the domain, vertical asymptote(s), and horizontal asymptote of a function include:

• Modeling population growth and decline with multiple variables.
• Modeling the spread of diseases with complex numbers.