Definition of the Function

Oct 13, 2025 by windailys.com 134 views

Introduction

The concept of a function is a fundamental aspect of mathematics, and it plays a crucial role in various branches of mathematics, including calculus, algebra, and analysis. A function is a relation between a set of inputs, called the domain, and a set of possible outputs, called the range. In this article, we will discuss a specific function $f$ defined on the interval $[-2, 2]$ , and we will explore its properties and behavior.

Definition of the Function

The function $f$ is defined on the interval $[-2, 2]$ as follows:

f(x) = \left\{ \begin{array}{ll} -2 & \text{if } -2 \leq x \ \textless \ -1 \\ -1 & \text{if } -1 \leq x \ \textless \ 0 \\ 0 & \text{if } 0 \leq x \ \textless \ 2 \\ \end{array} \right.

This function is a piecewise function, meaning that it is defined differently on different intervals. The function takes on the value $-2$ for $x$ in the interval $[-2, -1)$ , the value $-1$ for $x$ in the interval $[-1, 0)$ , and the value $0$ for $x$ in the interval $[0, 2)$ .

Properties of the Function

To understand the behavior of the function $f$ , we need to examine its properties. One of the most important properties of a function is its continuity. A function is continuous if it can be drawn without lifting the pencil from the paper. In other words, a function is continuous if its graph is a single, unbroken curve.

Let's examine the continuity of the function $f$ . We can see that the function is continuous on the intervals $[-2, -1)$ and $[0, 2)$ . However, the function is not continuous at the point $x = -1$ , because the left-hand limit and the right-hand limit of the function at this point are different.

Left-Hand Limit

To calculate the left-hand limit of the function $f$ at the point $x = -1$ , we need to examine the behavior of the function as $x$ approaches $-1$ from the left. In other words, we need to calculate the limit of the function as $x$ approaches $-1$ from the left.

\lim_{x \to -1^-} f(x) = \lim_{x \to -1^-} -2 = -2

This means that the left-hand limit of the function $f$ at the point $x = -1$ is $-2$ .

Right-Hand Limit

To calculate the right-hand limit of the function $f$ at the point $x = -1$ , we need to examine the behavior of the function as $x$ approaches $-1$ from the right. In other words, we need to calculate the limit of the function as $x$ approaches $-1$ from the right.

\lim_{x \to -1^+} f(x) = \lim_{x \to -1^+} -1 = -1

This means that the right-hand limit of the function $f$ at the point $x = -1$ is $-1$ .

Discontinuity

Since the left-hand limit and the right-hand limit of the function $f$ at the point $x = -1$ are different, the function is not continuous at this point. In other words, the function has a discontinuity at the point $x = -1$ .

Conclusion

In this article, we have discussed a specific function $f$ defined on the interval $[-2, 2]$ . We have examined the properties of the function, including its continuity and discontinuity. We have also calculated the left-hand limit and the right-hand limit of the function at the point $x = -1$ . Our results show that the function is not continuous at this point, and it has a discontinuity at the point $x = -1$ .

References

[1]: Calculus, by Michael Spivak.
[2]: Algebra, by Michael Artin.
[3]: Analysis, by Walter Rudin.

Glossary

Function: A relation between a set of inputs, called the domain, and a set of possible outputs, called the range.
Domain: The set of all possible inputs of a function.
Range: The set of all possible outputs of a function.
Continuity: A property of a function that means it can be drawn without lifting the pencil from the paper.
Discontinuity: A property of a function that means it cannot be drawn without lifting the pencil from the paper.
Left-hand limit: The limit of a function as $x$ approaches a point from the left.
Right-hand limit: The limit of a function as $x$ approaches a point from the right.
Suppose that the function $f$ is defined on the interval ${-2, 2$ }$ as follows: =====================================================

Q&A

In this section, we will answer some common questions related to the function $f$ defined on the interval $[-2, 2]$ .

Q: What is the value of the function $f$ at the point $x = -2$ ?

A: The value of the function $f$ at the point $x = -2$ is $-2$ .

Q: What is the value of the function $f$ at the point $x = -1$ ?

A: The value of the function $f$ at the point $x = -1$ is $-1$ .

Q: What is the value of the function $f$ at the point $x = 0$ ?

A: The value of the function $f$ at the point $x = 0$ is $0$ .

Q: Is the function $f$ continuous on the interval $[-2, 2]$ ?

A: No, the function $f$ is not continuous on the interval $[-2, 2]$ . It has a discontinuity at the point $x = -1$ .

Q: What is the left-hand limit of the function $f$ at the point $x = -1$ ?

A: The left-hand limit of the function $f$ at the point $x = -1$ is $-2$ .

Q: What is the right-hand limit of the function $f$ at the point $x = -1$ ?

A: The right-hand limit of the function $f$ at the point $x = -1$ is $-1$ .

Q: What is the domain of the function $f$ ?

A: The domain of the function $f$ is the interval $[-2, 2]$ .

Q: What is the range of the function $f$ ?

A: The range of the function $f$ is the set of all possible outputs of the function, which is $[-2, 0]$ .

Q: Is the function $f$ a one-to-one function?

A: No, the function $f$ is not a one-to-one function. It is a many-to-one function.

Q: What is the inverse of the function $f$ ?

A: The inverse of the function $f$ is not defined, because the function $f$ is not a one-to-one function.

Conclusion

In this article, we have answered some common questions related to the function $f$ defined on the interval $[-2, 2]$ . We have discussed the value of the function at different points, its continuity, left-hand and right-hand limits, domain, range, and inverse.

References

[1]: Calculus, by Michael Spivak.
[2]: Algebra, by Michael Artin.
[3]: Analysis, by Walter Rudin.

Glossary

Function: A relation between a set of inputs, called the domain, and a set of possible outputs, called the range.
Domain: The set of all possible inputs of a function.
Range: The set of all possible outputs of a function.
Continuity: A property of a function that means it can be drawn without lifting the pencil from the paper.
Discontinuity: A property of a function that means it cannot be drawn without lifting the pencil from the paper.
Left-hand limit: The limit of a function as $x$ approaches a point from the left.
Right-hand limit: The limit of a function as $x$ approaches a point from the right.
One-to-one function: A function that maps each input to a unique output.
Many-to-one function: A function that maps multiple inputs to the same output.
Inverse of a function: A function that undoes the action of the original function.