Let $f(x)=\frac{-9x+9}{-2x-5}$. Find The Domain, Vertical Asymptote(s), And Horizontal Asymptote.- Domain: $\square$ (express Your Answer Using Intervals)- Vertical Asymptote(s): $\square$ (enter Values As A Comma-separated

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Domain

To find the domain of the function $f(x)=\frac{-9x+9}{-2x-5}$, we need to determine the values of $x$ for which the denominator is not equal to zero. In other words, we need to find the values of $x$ that make the denominator $-2x-5$ equal to zero.

We can set the denominator equal to zero and solve for $x$:

$$-2x-5=0$$

Solving for $x$, we get:

$$x=-\frac{5}{2}$$

This means that the denominator is equal to zero when $x=-\frac{5}{2}$. Therefore, the domain of the function is all real numbers except $x=-\frac{5}{2}$.

In interval notation, the domain can be expressed as:

$$(-\infty, -\frac{5}{2}) \cup (-\frac{5}{2}, \infty)$$

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 $f(x)=\frac{-9x+9}{-2x-5}$, the vertical asymptote occurs when the denominator is equal to zero.

As we found earlier, the denominator is equal to zero when $x=-\frac{5}{2}$. Therefore, the vertical asymptote is the line $x=-\frac{5}{2}$.

Horizontal Asymptote

A horizontal asymptote is a horizontal line that the graph of a function approaches but never touches. In the case of the function $f(x)=\frac{-9x+9}{-2x-5}$, we can find the horizontal asymptote by looking at the degrees of the numerator and denominator.

The degree of the numerator is 1, and the degree of the denominator is 1. Since the degrees are the same, the horizontal asymptote is the ratio of the leading coefficients.

The leading coefficient of the numerator is -9, and the leading coefficient of the denominator is -2. Therefore, the horizontal asymptote is the line $y=\frac{-9}{-2}$.

Simplifying the ratio, we get:

$$y=\frac{9}{2}$$

Conclusion

In conclusion, the domain of the function $f(x)=\frac{-9x+9}{-2x-5}$ is all real numbers except $x=-\frac{5}{2}$. The vertical asymptote is the line $x=-\frac{5}{2}$, and the horizontal asymptote is the line $y=\frac{9}{2}$.

Step-by-Step Solution

Step 1: Find the Domain

To find the domain of the function, we need to determine the values of $x$ for which the denominator is not equal to zero.

We can set the denominator equal to zero and solve for $x$:

$$-2x-5=0$$

Solving for $x$, we get:

$$x=-\frac{5}{2}$$

This means that the denominator is equal to zero when $x=-\frac{5}{2}$. Therefore, the domain of the function is all real numbers except $x=-\frac{5}{2}$.

Step 2: Find the Vertical Asymptote

A vertical asymptote is a vertical line that the graph of a function approaches but never touches. In the case of the function $f(x)=\frac{-9x+9}{-2x-5}$, the vertical asymptote occurs when the denominator is equal to zero.

As we found earlier, the denominator is equal to zero when $x=-\frac{5}{2}$. Therefore, the vertical asymptote is the line $x=-\frac{5}{2}$.

Step 3: Find the Horizontal Asymptote

A horizontal asymptote is a horizontal line that the graph of a function approaches but never touches. In the case of the function $f(x)=\frac{-9x+9}{-2x-5}$, we can find the horizontal asymptote by looking at the degrees of the numerator and denominator.

The degree of the numerator is 1, and the degree of the denominator is 1. Since the degrees are the same, the horizontal asymptote is the ratio of the leading coefficients.

The leading coefficient of the numerator is -9, and the leading coefficient of the denominator is -2. Therefore, the horizontal asymptote is the line $y=\frac{-9}{-2}$.

Simplifying the ratio, we get:

$$y=\frac{9}{2}$$

Final Answer

The final answer is:

• Domain: $(-\infty, -\frac{5}{2}) \cup (-\frac{5}{2}, \infty)$
• Vertical Asymptote(s): $x=-\frac{5}{2}$
• Horizontal Asymptote: $y=\frac{9}{2}$
  

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Q: What is the domain of the function $f(x)=\frac{-9x+9}{-2x-5}$?

A: The domain of the function is all real numbers except $x=-\frac{5}{2}$. In interval notation, the domain can be expressed as:

$$(-\infty, -\frac{5}{2}) \cup (-\frac{5}{2}, \infty)$$

Q: Why is there a restriction on the domain of the function?

A: The restriction on the domain is due to the fact that the denominator of the function is equal to zero when $x=-\frac{5}{2}$. This means that the function is undefined at this point, and therefore, it is not included in the domain.

Q: What is the vertical asymptote of the function?

A: The vertical asymptote of the function is the line $x=-\frac{5}{2}$. This means that the graph of the function approaches but never touches this line.

Q: Why is there a vertical asymptote at $x=-\frac{5}{2}$?

A: There is a vertical asymptote at $x=-\frac{5}{2}$ because the denominator of the function is equal to zero at this point. This means that the function is undefined at this point, and therefore, the graph of the function approaches but never touches this line.

Q: What is the horizontal asymptote of the function?

A: The horizontal asymptote of the function is the line $y=\frac{9}{2}$. This means that the graph of the function approaches but never touches this line.

Q: Why is there a horizontal asymptote at $y=\frac{9}{2}$?

A: There is a horizontal asymptote at $y=\frac{9}{2}$ because the degrees of the numerator and denominator of the function are the same. This means that the ratio of the leading coefficients of the numerator and denominator is the horizontal asymptote.

Q: How do you find the horizontal asymptote of a rational function?

A: To find the horizontal asymptote of a rational function, you need to look at the degrees of the numerator and denominator. If the degrees are the same, the horizontal asymptote is the ratio of the leading coefficients. If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is the line $y=0$. 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 horizontal asymptote?

A: The horizontal asymptote is significant because it tells us the behavior of the function as $x$ approaches infinity. If the horizontal asymptote is a non-zero value, the function approaches that value as $x$ approaches infinity. If the horizontal asymptote is the line $y=0$, the function approaches zero as $x$ approaches infinity.

Q: How do you find the vertical asymptote of a rational function?

A: To find the vertical asymptote of a rational function, you need to set the denominator equal to zero and solve for $x$. The values of $x$ that make the denominator equal to zero are the vertical asymptotes.

Q: What is the significance of the vertical asymptote?

A: The vertical asymptote is significant because it tells us the behavior of the function as $x$ approaches the vertical asymptote. If the vertical asymptote is a non-zero value, the function approaches that value as $x$ approaches the vertical asymptote. If the vertical asymptote is the line $x=0$, the function approaches zero as $x$ approaches the vertical asymptote.

Q: Can a rational function have more than one vertical asymptote?

A: Yes, a rational function can have more than one vertical asymptote. This occurs when the denominator of the function is equal to zero at more than one point.

Q: Can a rational function have more than one horizontal asymptote?

A: No, a rational function cannot have more than one horizontal asymptote. The horizontal asymptote is determined by the ratio of the leading coefficients of the numerator and denominator, and this ratio is unique.

Q: How do you determine the domain of a rational function?

A: To determine the domain of a rational function, you need to find the values of $x$ that make the denominator equal to zero. These values are not included in the domain of the function.

Q: What is the significance of the domain of a rational function?

A: The domain of a rational function is significant because it tells us the values of $x$ for which the function is defined. If the domain is not all real numbers, the function is not defined for certain values of $x$.

Q: Can a rational function have a domain that is not all real numbers?

A: Yes, a rational function can have a domain that is not all real numbers. This occurs when the denominator of the function is equal to zero at certain points.

Q: How do you find the domain of a rational function with a denominator that is a quadratic expression?

A: To find the domain of a rational function with a denominator that is a quadratic expression, you need to factor the denominator and set each factor equal to zero. The values of $x$ that make each factor equal to zero are not included in the domain of the function.

Q: What is the significance of the domain of a rational function with a denominator that is a quadratic expression?

A: The domain of a rational function with a denominator that is a quadratic expression is significant because it tells us the values of $x$ for which the function is defined. If the domain is not all real numbers, the function is not defined for certain values of $x$.

Q: Can a rational function with a denominator that is a quadratic expression have a domain that is not all real numbers?

A: Yes, a rational function with a denominator that is a quadratic expression can have a domain that is not all real numbers. This occurs when the denominator is equal to zero at certain points.

Q: How do you find the domain of a rational function with a denominator that is a polynomial expression of degree greater than 2?

A: To find the domain of a rational function with a denominator that is a polynomial expression of degree greater than 2, you need to factor the denominator and set each factor equal to zero. The values of $x$ that make each factor equal to zero are not included in the domain of the function.

Q: What is the significance of the domain of a rational function with a denominator that is a polynomial expression of degree greater than 2?

A: The domain of a rational function with a denominator that is a polynomial expression of degree greater than 2 is significant because it tells us the values of $x$ for which the function is defined. If the domain is not all real numbers, the function is not defined for certain values of $x$.

Q: Can a rational function with a denominator that is a polynomial expression of degree greater than 2 have a domain that is not all real numbers?

A: Yes, a rational function with a denominator that is a polynomial expression of degree greater than 2 can have a domain that is not all real numbers. This occurs when the denominator is equal to zero at certain points.

Q: How do you find the domain of a rational function with a denominator that is a polynomial expression of degree 1?

A: To find the domain of a rational function with a denominator that is a polynomial expression of degree 1, you need to set the denominator equal to zero and solve for $x$. The values of $x$ that make the denominator equal to zero are not included in the domain of the function.

Q: What is the significance of the domain of a rational function with a denominator that is a polynomial expression of degree 1?

A: The domain of a rational function with a denominator that is a polynomial expression of degree 1 is significant because it tells us the values of $x$ for which the function is defined. If the domain is not all real numbers, the function is not defined for certain values of $x$.

Q: Can a rational function with a denominator that is a polynomial expression of degree 1 have a domain that is not all real numbers?

A: Yes, a rational function with a denominator that is a polynomial expression of degree 1 can have a domain that is not all real numbers. This occurs when the denominator is equal to zero at certain points.

Q: How do you find the domain of a rational function with a denominator that is a polynomial expression of degree 0?

A: To find the domain of a rational function with a denominator that is a polynomial expression of degree 0, you need to set the denominator equal to zero and solve for $x$. The values of $x$ that make the denominator equal to zero are not included in the domain of the function.

Q: What is the significance of the domain of a rational function with a denominator that is a polynomial expression of degree 0?

A: The domain of a rational function with a denominator that is a polynomial expression of degree 0 is significant because it tells us the values of $x$ for which the function is defined. If the domain is not all real numbers, the function is not defined for certain values of $x$.

Q: Can a rational function with a denominator that is a polynomial expression of degree 0 have a domain that is not all real numbers?

A: Yes, a rational function with a denominator that is a polynomial expression of degree 0 can have a domain that is not all real numbers. This occurs when the