Determine Whether The Given Piecewise Function Is Continuous At The Given Value Of $a$. If It Is Discontinuous, Identify The Type Of Discontinuity.Part A) $\[ g(x) = \begin{cases} 1-x, & \text{if } X \ \textless \ -4 \\ x^2 - 11, &

Introduction

In mathematics, a function is considered continuous if its graph can be drawn without lifting the pencil from the paper. In other words, a function is continuous if its graph has no gaps or jumps. Piecewise functions are a type of function that is defined by multiple sub-functions, each applied to a specific interval of the domain. In this article, we will determine whether a given piecewise function is continuous at a specific value of $a$ and identify the type of discontinuity if it is not continuous.

What is Continuity?

A function $f(x)$ is said to be continuous at a point $x = a$ if the following conditions are met:

1. The function is defined at $x = a$, i.e., $f(a)$ is defined.
2. The limit of the function as $x$ approaches $a$ exists, i.e., $\lim_{x \to a} f(x)$ exists.
3. The limit of the function as $x$ approaches $a$ is equal to the value of the function at $x = a$, i.e., $\lim_{x \to a} f(x) = f(a)$.

Types of Discontinuity

If a function is not continuous at a point $x = a$, it is said to be discontinuous at that point. There are three types of discontinuity:

1. Removable Discontinuity: A removable discontinuity occurs when the limit of the function as $x$ approaches $a$ exists, but the function is not defined at $x = a$. In this case, the function can be made continuous by assigning a value to the function at $x = a$.
2. Jump Discontinuity: A jump discontinuity occurs when the limit of the function as $x$ approaches $a$ from the left and right exist, but are not equal. In this case, the function has a jump or a gap at $x = a$.
3. Infinite Discontinuity: An infinite discontinuity occurs when the limit of the function as $x$ approaches $a$ does not exist because the function approaches infinity or negative infinity.

Piecewise Function

The given piecewise function is:

$$g(x) = \begin{cases} 1-x, & \text{if } x < -4 \\ x^2 - 11, & \text{if } x \geq -4 \end{cases}$$

We need to determine whether this function is continuous at $x = -4$.

Step 1: Check if the function is defined at $x = -4$

The function $g(x)$ is defined as $1-x$ if $x < -4$ and $x^2 - 11$ if $x \geq -4$. Since $x = -4$ is not less than $-4$, the function $g(x)$ is defined as $x^2 - 11$ at $x = -4$. Therefore, the function is defined at $x = -4$.

Step 2: Check if the limit of the function as $x$ approaches $-4$ exists

We need to check if the limit of the function as $x$ approaches $-4$ from the left and right exist.

Left-hand limit

The left-hand limit of the function as $x$ approaches $-4$ is:

$$\lim_{x \to -4^-} g(x) = \lim_{x \to -4^-} (1-x)$$

Using the limit properties, we can rewrite this as:

$$\lim_{x \to -4^-} g(x) = \lim_{x \to -4^-} 1 - \lim_{x \to -4^-} x$$

Since the limit of $1$ as $x$ approaches $-4$ is $1$, and the limit of $x$ as $x$ approaches $-4$ from the left is $-4$, we have:

$$\lim_{x \to -4^-} g(x) = 1 - (-4) = 5$$

Right-hand limit

The right-hand limit of the function as $x$ approaches $-4$ is:

$$\lim_{x \to -4^+} g(x) = \lim_{x \to -4^+} (x^2 - 11)$$

Using the limit properties, we can rewrite this as:

$$\lim_{x \to -4^+} g(x) = \lim_{x \to -4^+} x^2 - \lim_{x \to -4^+} 11$$

Since the limit of $x^2$ as $x$ approaches $-4$ from the right is $16$, and the limit of $11$ as $x$ approaches $-4$ from the right is $11$, we have:

$$\lim_{x \to -4^+} g(x) = 16 - 11 = 5$$

Conclusion

Since the left-hand and right-hand limits of the function as $x$ approaches $-4$ exist and are equal, the limit of the function as $x$ approaches $-4$ exists. Therefore, the function $g(x)$ is continuous at $x = -4$.

Final Answer

The given piecewise function $g(x)$ is continuous at $x = -4$.

Discussion

In this article, we determined whether a given piecewise function is continuous at a specific value of $a$ and identified the type of discontinuity if it is not continuous. We used the definition of continuity and the properties of limits to analyze the function and determine its continuity at $x = -4$. The function $g(x)$ is continuous at $x = -4$ because the left-hand and right-hand limits of the function as $x$ approaches $-4$ exist and are equal.

References

• [1] Thomas, G. B. (2014). Calculus and Analytic Geometry. Pearson Education.
• [2] Larson, R. E. (2015). Calculus: Early Transcendentals. Cengage Learning.
• [3] Stewart, J. (2015). Calculus: Early Transcendentals. Cengage Learning.

Keywords

• Piecewise function
• Continuity
• Discontinuity
• Removable discontinuity
• Jump discontinuity
• Infinite discontinuity
• Limit
• Left-hand limit
• Right-hand limit
• Calculus
• Analytic geometry
  Q&A: Piecewise Functions and Continuity =============================================

Introduction

In our previous article, we discussed the concept of continuity in piecewise functions and determined whether a given piecewise function is continuous at a specific value of $a$. In this article, we will answer some frequently asked questions about piecewise functions and continuity.

Q: What is a piecewise function?

A piecewise function is a type of function that is defined by multiple sub-functions, each applied to a specific interval of the domain.

Q: What is continuity in a function?

A function is considered continuous if its graph can be drawn without lifting the pencil from the paper. In other words, a function is continuous if its graph has no gaps or jumps.

Q: How do I determine if a piecewise function is continuous at a specific value of $a$?

To determine if a piecewise function is continuous at a specific value of $a$, you need to check if the function is defined at $x = a$, if the limit of the function as $x$ approaches $a$ exists, and if the limit of the function as $x$ approaches $a$ is equal to the value of the function at $x = a$.

Q: What are the types of discontinuity in a function?

There are three types of discontinuity in a function:

1. Removable Discontinuity: A removable discontinuity occurs when the limit of the function as $x$ approaches $a$ exists, but the function is not defined at $x = a$.
2. Jump Discontinuity: A jump discontinuity occurs when the limit of the function as $x$ approaches $a$ from the left and right exist, but are not equal.
3. Infinite Discontinuity: An infinite discontinuity occurs when the limit of the function as $x$ approaches $a$ does not exist because the function approaches infinity or negative infinity.

Q: How do I identify the type of discontinuity in a function?

To identify the type of discontinuity in a function, you need to check if the limit of the function as $x$ approaches $a$ exists and if it is equal to the value of the function at $x = a$. If the limit exists but is not equal to the value of the function at $x = a$, then the function has a removable discontinuity. If the limit exists but is not equal to the value of the function at $x = a$ and the function has a jump or a gap at $x = a$, then the function has a jump discontinuity. If the limit does not exist because the function approaches infinity or negative infinity, then the function has an infinite discontinuity.

Q: Can a piecewise function have multiple discontinuities?

Yes, a piecewise function can have multiple discontinuities. Each sub-function in the piecewise function can have its own discontinuities.

Q: How do I graph a piecewise function?

To graph a piecewise function, you need to graph each sub-function separately and then combine them to form the graph of the piecewise function.

Q: What are some common applications of piecewise functions?

Piecewise functions have many applications in mathematics, science, and engineering. Some common applications include:

1. Modeling real-world phenomena: Piecewise functions can be used to model real-world phenomena such as population growth, temperature changes, and economic trends.
2. Solving optimization problems: Piecewise functions can be used to solve optimization problems such as finding the maximum or minimum value of a function.
3. Analyzing data: Piecewise functions can be used to analyze data and identify patterns and trends.

Conclusion

In this article, we answered some frequently asked questions about piecewise functions and continuity. We discussed the concept of continuity in piecewise functions, the types of discontinuity, and how to identify the type of discontinuity in a function. We also discussed some common applications of piecewise functions.

Final Answer

The final answer to the question of whether a piecewise function is continuous at a specific value of $a$ depends on the specific function and the value of $a$. However, by following the steps outlined in this article, you can determine if a piecewise function is continuous at a specific value of $a$ and identify the type of discontinuity if it is not continuous.

References

• [1] Thomas, G. B. (2014). Calculus and Analytic Geometry. Pearson Education.
• [2] Larson, R. E. (2015). Calculus: Early Transcendentals. Cengage Learning.
• [3] Stewart, J. (2015). Calculus: Early Transcendentals. Cengage Learning.

Keywords

• Piecewise function
• Continuity
• Discontinuity
• Removable discontinuity
• Jump discontinuity
• Infinite discontinuity
• Limit
• Left-hand limit
• Right-hand limit
• Calculus
• Analytic geometry