Given $g(x)=\frac{3x^2+12x}{x^3+2x^2-8x}$, Do The Following. Show All Work.a. Find All Asymptotes, Holes, And Zeros.b. State The Domain Of The Function.c. Use Limits To Describe The Behavior At The Vertical Asymptotes And The End Behavior.

Introduction

In this article, we will explore the properties of a given rational function, specifically its asymptotes, holes, and zeros. We will also determine the domain of the function and use limits to describe its behavior at the vertical asymptotes and the end behavior.

Step 1: Factor the numerator and denominator

To begin, we need to factor the numerator and denominator of the given rational function.

$$g(x)=\frac{3x^2+12x}{x^3+2x^2-8x}$$

We can factor out a common factor of $x$ from both the numerator and denominator:

$$g(x)=\frac{x(3x+12)}{x(x^2+2x-8)}$$

Now, we can factor the quadratic expression in the denominator:

$$g(x)=\frac{x(3x+12)}{x(x+4)(x-2)}$$

Step 2: Find the zeros of the function

To find the zeros of the function, we need to set the numerator equal to zero and solve for $x$.

$$x(3x+12)=0$$

This gives us two possible solutions:

$$x=0 \quad \text{or} \quad 3x+12=0$$

Solving for the second solution, we get:

$$3x=-12 \quad \text{implies} \quad x=-4$$

Therefore, the zeros of the function are $x=0$ and $x=-4$.

Step 3: Find the vertical asymptotes

To find the vertical asymptotes, we need to set the denominator equal to zero and solve for $x$.

$$x(x+4)(x-2)=0$$

This gives us three possible solutions:

$$x=0 \quad \text{or} \quad x+4=0 \quad \text{or} \quad x-2=0$$

Solving for the second and third solutions, we get:

$$x=-4 \quad \text{or} \quad x=2$$

Therefore, the vertical asymptotes are $x=0$, $x=-4$, and $x=2$.

Step 4: Find the holes

To find the holes, we need to check if there are any common factors between the numerator and denominator.

In this case, we can see that there is a common factor of $x$ between the numerator and denominator.

Therefore, the hole is $x=0$.

Step 5: Determine the domain of the function

The domain of the function is all real numbers except for the values that make the denominator equal to zero.

In this case, the denominator is equal to zero when $x=0$, $x=-4$, or $x=2$.

Therefore, the domain of the function is:

$$x \in (-\infty, -4) \cup (-4, 0) \cup (0, 2) \cup (2, \infty)$$

Step 6: Use limits to describe the behavior at the vertical asymptotes

To describe the behavior at the vertical asymptotes, we can use limits.

For the vertical asymptote $x=0$, we can evaluate the limit as $x$ approaches $0$ from the left and right:

$$\lim_{x \to 0^-} g(x) = \lim_{x \to 0^+} g(x) = \infty$$

This indicates that the function approaches positive infinity as $x$ approaches $0$ from the left and right.

For the vertical asymptote $x=-4$, we can evaluate the limit as $x$ approaches $-4$ from the left and right:

$$\lim_{x \to -4^-} g(x) = \lim_{x \to -4^+} g(x) = -\infty$$

This indicates that the function approaches negative infinity as $x$ approaches $-4$ from the left and right.

For the vertical asymptote $x=2$, we can evaluate the limit as $x$ approaches $2$ from the left and right:

$$\lim_{x \to 2^-} g(x) = \lim_{x \to 2^+} g(x) = \infty$$

This indicates that the function approaches positive infinity as $x$ approaches $2$ from the left and right.

Step 7: Use limits to describe the end behavior

To describe the end behavior, we can evaluate the limit as $x$ approaches positive or negative infinity.

For the function $g(x)$, we can evaluate the limit as $x$ approaches positive or negative infinity:

$$\lim_{x \to \infty} g(x) = \lim_{x \to -\infty} g(x) = 0$$

This indicates that the function approaches $0$ as $x$ approaches positive or negative infinity.

Conclusion

In this article, we have explored the properties of a given rational function, specifically its asymptotes, holes, and zeros. We have also determined the domain of the function and used limits to describe its behavior at the vertical asymptotes and the end behavior.

The zeros of the function are $x=0$ and $x=-4$.

The vertical asymptotes are $x=0$, $x=-4$, and $x=2$.

The hole is $x=0$.

The domain of the function is:

$$x \in (-\infty, -4) \cup (-4, 0) \cup (0, 2) \cup (2, \infty)$$

The function approaches positive infinity as $x$ approaches $0$ from the left and right.

The function approaches negative infinity as $x$ approaches $-4$ from the left and right.

The function approaches positive infinity as $x$ approaches $2$ from the left and right.

The function approaches $0$ as $x$ approaches positive or negative infinity.

By understanding the properties of this rational function, we can better analyze and solve problems involving rational functions.

Introduction

In our previous article, we explored the properties of a given rational function, specifically its asymptotes, holes, and zeros. We also determined the domain of the function and used limits to describe its behavior at the vertical asymptotes and the end behavior. In this article, we will answer some frequently asked questions about rational functions and provide additional insights into the properties of these functions.

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

A: A vertical asymptote is a value of x that makes the denominator of the rational function equal to zero, resulting in an infinite or undefined value. A hole, on the other hand, is a value of x that makes both the numerator and denominator equal to zero, resulting in a removable discontinuity.

Q: How do I 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 called vertical asymptotes, and they must be excluded from the domain.

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

A: The zeros of a rational function are the values of x that make the numerator equal to zero. These values are important because they can affect the behavior of the function, particularly at the vertical asymptotes.

Q: How do I use limits to describe the behavior of a rational function at a vertical asymptote?

A: To describe the behavior of a rational function at a vertical asymptote, you can use limits. Specifically, you can evaluate the limit of the function as x approaches the vertical asymptote from the left and right. This will give you an idea of whether the function approaches positive or negative infinity.

Q: What is the end behavior of a rational function?

A: The end behavior of a rational function refers to the behavior of the function as x approaches positive or negative infinity. This can be determined by evaluating the limit of the function as x approaches positive or negative infinity.

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 has multiple factors that make it equal to zero.

Q: How do I determine the degree of a rational function?

A: The degree of a rational function is the highest power of x in the numerator or denominator. This can be determined by examining the terms of the numerator and denominator.

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

A: The degree of a rational function is important because it can affect the behavior of the function, particularly at the vertical asymptotes. A rational function with a higher degree can have more complex behavior.

Q: Can a rational function have a hole and a vertical asymptote at the same value of x?

A: Yes, a rational function can have a hole and a vertical asymptote at the same value of x. This occurs when the numerator and denominator have a common factor that makes both equal to zero.

Q: How do I graph a rational function?

A: To graph a rational function, you can start by plotting the zeros and vertical asymptotes of the function. Then, you can use the behavior of the function at these points to determine the overall shape of the graph.

Q: What are some common mistakes to avoid when working with rational functions?

A: Some common mistakes to avoid when working with rational functions include:

• Failing to factor the numerator and denominator
• Failing to identify the zeros and vertical asymptotes of the function
• Failing to use limits to describe the behavior of the function at the vertical asymptotes
• Failing to consider the degree of the function when determining its behavior

By understanding these common mistakes, you can avoid them and ensure that your work with rational functions is accurate and reliable.

Conclusion

In this article, we have answered some frequently asked questions about rational functions and provided additional insights into the properties of these functions. We have also discussed some common mistakes to avoid when working with rational functions. By understanding these concepts, you can better analyze and solve problems involving rational functions.