What Are The Coordinates Of The Removable Discontinuity Of The Function $f(x)=\frac{(x-3)(x-2)}{x(x-3)(x+1)}$?

Introduction

Removable discontinuities are a type of discontinuity that occurs in a function when the function is not defined at a particular point, but the limit of the function as it approaches that point exists. In other words, a removable discontinuity occurs when a function has a hole or a gap in its graph. The function $f(x)=\frac{(x-3)(x-2)}{x(x-3)(x+1)}$ has a removable discontinuity, and in this article, we will determine the coordinates of this discontinuity.

Understanding the Function

The given function is a rational function, which means it is the ratio of two polynomials. The numerator of the function is $(x-3)(x-2)$, and the denominator is $x(x-3)(x+1)$. To find the removable discontinuity, we need to find the values of $x$ for which the function is not defined.

Finding the Discontinuity

A rational function is not defined when the denominator is equal to zero. Therefore, we need to find the values of $x$ for which the denominator $x(x-3)(x+1)$ is equal to zero. We can do this by setting the denominator equal to zero and solving for $x$.

$$x(x-3)(x+1) = 0$$

This equation has three solutions: $x=0$, $x=3$, and $x=-1$. However, we need to check if these solutions are removable discontinuities.

Checking for Removable Discontinuities

A removable discontinuity occurs when the limit of the function as it approaches a particular point exists, but the function is not defined at that point. To check if a solution is a removable discontinuity, we need to check if the limit of the function as it approaches that point exists.

Let's check the solution $x=3$. We can do this by factoring the numerator and the denominator of the function.

$$f(x)=\frac{(x-3)(x-2)}{x(x-3)(x+1)} = \frac{(x-2)}{x(x+1)}$$

Now, we can see that the function is not defined at $x=3$, but the limit of the function as it approaches $x=3$ exists. Therefore, $x=3$ is a removable discontinuity.

Finding the Coordinates of the Discontinuity

The coordinates of a removable discontinuity are the values of $x$ and $y$ at which the function is not defined. In this case, the coordinates of the removable discontinuity are $(3, f(3))$.

To find the value of $f(3)$, we can substitute $x=3$ into the function.

$$f(3) = \frac{(3-2)}{3(3+1)} = \frac{1}{12}$$

Therefore, the coordinates of the removable discontinuity are $(3, \frac{1}{12})$.

Conclusion

In this article, we determined the coordinates of the removable discontinuity of the function $f(x)=\frac{(x-3)(x-2)}{x(x-3)(x+1)}$. We found that the function has a removable discontinuity at $x=3$, and the coordinates of this discontinuity are $(3, \frac{1}{12})$.

Step-by-Step Solution

Step 1: Factor the numerator and the denominator of the function

$$f(x)=\frac{(x-3)(x-2)}{x(x-3)(x+1)} = \frac{(x-2)}{x(x+1)}$$

Step 2: Check if the limit of the function as it approaches $x=3$ exists

The limit of the function as it approaches $x=3$ exists, but the function is not defined at $x=3$. Therefore, $x=3$ is a removable discontinuity.

Step 3: Find the value of $f(3)$

$$f(3) = \frac{(3-2)}{3(3+1)} = \frac{1}{12}$$

Step 4: Determine the coordinates of the removable discontinuity

The coordinates of the removable discontinuity are $(3, \frac{1}{12})$.

Frequently Asked Questions

Q: What is a removable discontinuity?

A: A removable discontinuity is a type of discontinuity that occurs in a function when the function is not defined at a particular point, but the limit of the function as it approaches that point exists.

Q: How do I find the removable discontinuity of a function?

A: To find the removable discontinuity of a function, you need to find the values of $x$ for which the function is not defined. Then, you need to check if the limit of the function as it approaches those points exists.

Q: What are the coordinates of the removable discontinuity?

A: The coordinates of the removable discontinuity are the values of $x$ and $y$ at which the function is not defined.

Further Reading

• Removable Discontinuity
• Limits
• Rational Functions
  

Introduction

In our previous article, we discussed the concept of removable discontinuity and how to find the coordinates of a removable discontinuity. In this article, we will answer some frequently asked questions about removable discontinuity.

Q&A

Q: What is a removable discontinuity?

A: A removable discontinuity is a type of discontinuity that occurs in a function when the function is not defined at a particular point, but the limit of the function as it approaches that point exists.

Q: How do I know if a function has a removable discontinuity?

A: To determine if a function has a removable discontinuity, you need to check if the function is not defined at a particular point, but the limit of the function as it approaches that point exists.

Q: How do I find the removable discontinuity of a function?

A: To find the removable discontinuity of a function, you need to find the values of $x$ for which the function is not defined. Then, you need to check if the limit of the function as it approaches those points exists.

Q: What are the coordinates of the removable discontinuity?

A: The coordinates of the removable discontinuity are the values of $x$ and $y$ at which the function is not defined.

Q: Can a removable discontinuity occur at a point where the function is undefined?

A: Yes, a removable discontinuity can occur at a point where the function is undefined. However, the limit of the function as it approaches that point must exist.

Q: Can a removable discontinuity occur at a point where the function is defined?

A: No, a removable discontinuity cannot occur at a point where the function is defined.

Q: How do I determine if a removable discontinuity is a hole or a gap in the graph of the function?

A: To determine if a removable discontinuity is a hole or a gap in the graph of the function, you need to check if the function is continuous at the point of the removable discontinuity.

Q: Can a removable discontinuity occur at a point where the function is continuous?

A: No, a removable discontinuity cannot occur at a point where the function is continuous.

Q: How do I find the value of the function at a removable discontinuity?

A: To find the value of the function at a removable discontinuity, you need to substitute the value of $x$ into the function.

Q: Can a removable discontinuity occur at a point where the function is not continuous?

A: Yes, a removable discontinuity can occur at a point where the function is not continuous.

Examples

Example 1: Find the removable discontinuity of the function $f(x)=\frac{(x-3)(x-2)}{x(x-3)(x+1)}$

The function has a removable discontinuity at $x=3$, and the coordinates of this discontinuity are $(3, \frac{1}{12})$.

Example 2: Find the removable discontinuity of the function $f(x)=\frac{x^2-4}{x(x-2)}$

The function has a removable discontinuity at $x=2$, and the coordinates of this discontinuity are $(2, 2)$.

Conclusion

In this article, we answered some frequently asked questions about removable discontinuity. We discussed how to determine if a function has a removable discontinuity, how to find the removable discontinuity of a function, and how to find the coordinates of a removable discontinuity.

Step-by-Step Solution

Step 1: Check if the function is not defined at a particular point

To determine if a function has a removable discontinuity, you need to check if the function is not defined at a particular point.

Step 2: Check if the limit of the function as it approaches the point exists

If the function is not defined at a particular point, you need to check if the limit of the function as it approaches that point exists.

Step 3: Find the value of the function at the removable discontinuity

To find the value of the function at a removable discontinuity, you need to substitute the value of $x$ into the function.

Step 4: Determine the coordinates of the removable discontinuity

The coordinates of the removable discontinuity are the values of $x$ and $y$ at which the function is not defined.

Frequently Asked Questions

Q: What is a removable discontinuity?

A: A removable discontinuity is a type of discontinuity that occurs in a function when the function is not defined at a particular point, but the limit of the function as it approaches that point exists.

Q: How do I know if a function has a removable discontinuity?

A: To determine if a function has a removable discontinuity, you need to check if the function is not defined at a particular point, but the limit of the function as it approaches that point exists.

Q: How do I find the removable discontinuity of a function?

A: To find the removable discontinuity of a function, you need to find the values of $x$ for which the function is not defined. Then, you need to check if the limit of the function as it approaches those points exists.

Q: What are the coordinates of the removable discontinuity?

A: The coordinates of the removable discontinuity are the values of $x$ and $y$ at which the function is not defined.

Further Reading

• Removable Discontinuity
• Limits
• Rational Functions