Determining the Type of Discontinuity

Oct 13, 2025 by windailys.com 3 views

What is the Discontinuity of the Function $f(x) = \frac{x^2 - 4x - 12}{x + 2}$ ?

In mathematics, a function is a relation between a set of inputs, called the domain, and a set of possible outputs, called the range. A function can be represented graphically as a curve on a coordinate plane. However, not all functions are continuous, meaning they do not have any breaks or gaps in their graph. In this article, we will explore the concept of discontinuity in functions and determine the discontinuity of the function $f(x) = \frac{x^2 - 4x - 12}{x + 2}$ .

Discontinuity occurs when a function has a break or gap in its graph, meaning that the function is not defined at a particular point or set of points. This can happen for several reasons, including:

Division by zero: When a function involves division by zero, it is undefined at that point.
Undefined limits: When the limit of a function approaches a certain value, but the function is not defined at that point.
Jump discontinuities: When a function has a sudden change in its value at a particular point.
Infinite discontinuities: When a function has an infinite value at a particular point.

The Function $f(x) = \frac{x^2 - 4x - 12}{x + 2}$

The given function is a rational function, which means it is the ratio of two polynomials. The numerator is $x^2 - 4x - 12$ , and the denominator is $x + 2$ . To determine the discontinuity of this function, we need to examine the denominator, as division by zero is undefined.

Finding the Discontinuity

To find the discontinuity of the function, we need to find the value of $x$ that makes the denominator equal to zero. We can do this by setting the denominator equal to zero and solving for $x$ :

$x + 2 = 0$

Subtracting 2 from both sides gives us:

$x = -2$

Therefore, the function is undefined at $x = -2$ , which means that the discontinuity of the function is at $x = -2$ .

Determining the Type of Discontinuity

To determine the type of discontinuity, we need to examine the behavior of the function as $x$ approaches $-2$ . We can do this by using limits:

$\lim_{x \to -2} \frac{x^2 - 4x - 12}{x + 2}$

Using algebraic manipulation, we can simplify the expression:

$\lim_{x \to -2} \frac{(x + 2)(x - 6)}{x + 2}$

Canceling out the common factor of $x + 2$ , we get:

$\lim_{x \to -2} (x - 6)$

Evaluating the limit, we get:

$-2 - 6 = -8$

Therefore, the function approaches $-8$ as $x$ approaches $-2$ . This means that the discontinuity is a jump discontinuity, as the function has a sudden change in its value at $x = -2$ .

In conclusion, the discontinuity of the function $f(x) = \frac{x^2 - 4x - 12}{x + 2}$ is at $x = -2$ , which is a jump discontinuity. This means that the function is not defined at $x = -2$ , and the graph of the function has a break or gap at that point.

The correct answer is:

C. ${$ (-2, -8)$}$

$This answer indicates that the discontinuity of the function is at $x = -2$, and the function approaches $-8$ as $x$ approaches $-2$.<br/> **Q&A: Discontinuity of the Function $f(x) = \frac{x^2 - 4x - 12}{x + 2}$**$

In our previous article, we explored the concept of discontinuity in functions and determined the discontinuity of the function $f(x) = \frac{x^2 - 4x - 12}{x + 2}$ . In this article, we will answer some frequently asked questions about the discontinuity of this function.

Q: What is the definition of discontinuity in a function?

A: Discontinuity in a function occurs when the function is not defined at a particular point or set of points. This can happen for several reasons, including division by zero, undefined limits, jump discontinuities, and infinite discontinuities.

Q: Why is the function $f(x) = \frac{x^2 - 4x - 12}{x + 2}$ undefined at $x = -2$ ?

A: The function is undefined at $x = -2$ because the denominator is equal to zero at that point. Division by zero is undefined, so the function is not defined at $x = -2$ .

Q: What type of discontinuity is the function $f(x) = \frac{x^2 - 4x - 12}{x + 2}$ at $x = -2$ ?

A: The function is a jump discontinuity at $x = -2$ . This means that the function has a sudden change in its value at that point.

Q: How do you determine the type of discontinuity in a function?

A: To determine the type of discontinuity in a function, you need to examine the behavior of the function as it approaches the point of discontinuity. You can do this by using limits.

Q: What is the limit of the function $f(x) = \frac{x^2 - 4x - 12}{x + 2}$ as $x$ approaches $-2$ ?

A: The limit of the function as $x$ approaches $-2$ is $-8$ . This means that the function approaches $-8$ as $x$ approaches $-2$ .

Q: Why is the limit of the function $f(x) = \frac{x^2 - 4x - 12}{x + 2}$ as $x$ approaches $-2$ equal to $-8$ ?

A: The limit of the function as $x$ approaches $-2$ is equal to $-8$ because the function can be simplified to $(x - 6)$ , and the limit of $(x - 6)$ as $x$ approaches $-2$ is $-8$ .

Q: What is the significance of the discontinuity of the function $f(x) = \frac{x^2 - 4x - 12}{x + 2}$ at $x = -2$ ?

A: The discontinuity of the function at $x = -2$ is significant because it means that the function is not defined at that point. This can affect the behavior of the function and its graph.

Q: What is the definition of discontinuity in a function?
A: Discontinuity in a function occurs when the function is not defined at a particular point or set of points.
Q: Why is the function $f(x) = \frac{x^2 - 4x - 12}{x + 2}$ undefined at $x = -2$ ?
A: The function is undefined at $x = -2$ because the denominator is equal to zero at that point.
Q: What type of discontinuity is the function $f(x) = \frac{x^2 - 4x - 12}{x + 2}$ at $x = -2$ ?
A: The function is a jump discontinuity at $x = -2$ .

For more information on discontinuity in functions, see our previous article on the topic.
For more information on limits, see our article on limits.
For more information on the function $f(x) = \frac{x^2 - 4x - 12}{x + 2}$ , see our article on the function.