It is a subset of the codomain

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Introduction

In mathematics, the domain of a function is the set of all possible input values for which the function is defined. In other words, it is the set of all possible values of the variable (in this case, x) that can be plugged into the function without causing any mathematical inconsistencies or undefined results. In this article, we will explore the concept of the domain of a function and find the domain of the given function $u(x) = \sqrt{-x} - 5$ .

What is the Domain of a Function?

The domain of a function is a crucial concept in mathematics, as it determines the set of all possible input values for which the function is defined. In other words, it is the set of all possible values of the variable (in this case, x) that can be plugged into the function without causing any mathematical inconsistencies or undefined results.

Properties of the Domain

The domain of a function has several important properties:

It is a set: The domain of a function is a set of all possible input values for which the function is defined.
It is a subset of the codomain: The domain of a function is a subset of the codomain, which is the set of all possible output values of the function.
It is a non-empty set: The domain of a function is a non-empty set, meaning that it contains at least one element.

Finding the Domain of a Function

To find the domain of a function, we need to identify the set of all possible input values for which the function is defined. In other words, we need to find the set of all possible values of the variable (in this case, x) that can be plugged into the function without causing any mathematical inconsistencies or undefined results.

Step 1: Identify the Type of Function

The given function $u(x) = \sqrt{-x} - 5$ is a square root function. The square root function is defined only for non-negative real numbers.

Step 2: Identify the Restrictions on the Variable

The given function $u(x) = \sqrt{-x} - 5$ has a restriction on the variable x, which is that x must be less than or equal to 0.

Step 3: Write the Domain in Interval Notation

The domain of the function $u(x) = \sqrt{-x} - 5$ can be written in interval notation as $(-\infty, 0]$ .

Conclusion

In conclusion, the domain of the function $u(x) = \sqrt{-x} - 5$ is the set of all possible input values for which the function is defined. The domain of the function can be written in interval notation as $(-\infty, 0]$ .

Final Answer

The final answer is $\boxed{(-\infty, 0]}$ .

Domain of a Function: Understanding the Concept

Introduction

What is the Domain of a Function?

Properties of the Domain

The domain of a function has several important properties:

It is a set: The domain of a function is a set of all possible input values for which the function is defined.
It is a subset of the codomain: The domain of a function is a subset of the codomain, which is the set of all possible output values of the function.
It is a non-empty set: The domain of a function is a non-empty set, meaning that it contains at least one element.

Finding the Domain of a Function

Step 1: Identify the Type of Function

The given function $u(x) = \sqrt{-x} - 5$ is a square root function. The square root function is defined only for non-negative real numbers.

Step 2: Identify the Restrictions on the Variable

The given function $u(x) = \sqrt{-x} - 5$ has a restriction on the variable x, which is that x must be less than or equal to 0.

Step 3: Write the Domain in Interval Notation

The domain of the function $u(x) = \sqrt{-x} - 5$ can be written in interval notation as $(-\infty, 0]$ .

Conclusion

Final Answer

Q&A: Domain of a Function

Q: What is the domain of a function?

A: The domain of a function is the set of all possible input values for which the function is defined. In other words, it is the set of all possible values of the variable (in this case, x) that can be plugged into the function without causing any mathematical inconsistencies or undefined results.

Q: Why is the domain of a function important?

A: The domain of a function is important because it determines the set of all possible input values for which the function is defined. This is crucial in mathematics, as it helps us to understand the behavior of the function and to make predictions about its output.

Q: How do I find the domain of a function?

A: To find the domain of a function, you need to identify the set of all possible input values for which the function is defined. This involves analyzing the function and identifying any restrictions on the variable.

Q: What are some common restrictions on the variable?

A: Some common restrictions on the variable include:

Non-negative real numbers: The square root function is defined only for non-negative real numbers.
Positive real numbers: The logarithmic function is defined only for positive real numbers.
Real numbers: The polynomial function is defined for all real numbers.

Q: How do I write the domain in interval notation?

A: To write the domain in interval notation, you need to identify the set of all possible input values for which the function is defined. This involves using interval notation to represent the domain.

Q: What is interval notation?

A: Interval notation is a way of representing a set of numbers using a specific notation. It is commonly used to represent the domain of a function.

Q: How do I use interval notation to represent the domain?

A: To use interval notation to represent the domain, you need to identify the set of all possible input values for which the function is defined. This involves using the following notation:

(-∞, a]: The domain is all real numbers less than or equal to a.
[a, ∞): The domain is all real numbers greater than or equal to a.
(-∞, a) ∪ (b, ∞): The domain is all real numbers less than or equal to a and all real numbers greater than or equal to b.

Q: Can you give an example of how to use interval notation to represent the domain?

A: Yes, let's consider the function $u(x) = \sqrt{-x} - 5$ . The domain of this function is all real numbers less than or equal to 0. To represent this in interval notation, we would use the following notation:

$(-\infty, 0]$

Q: What is the final answer?

A: The final answer is $\boxed{(-\infty, 0]}$ .

Domain of a Function: Understanding the Concept

Introduction

What is the Domain of a Function?

Properties of the Domain

The domain of a function has several important properties:

It is a set: The domain of a function is a set of all possible input values for which the function is defined.
It is a subset of the codomain: The domain of a function is a subset of the codomain, which is the set of all possible output values of the function.
It is a non-empty set: The domain of a function is a non-empty set, meaning that it contains at least one element.

Finding the Domain of a Function

Step 1: Identify the Type of Function

The given function $u(x) = \sqrt{-x} - 5$ is a square root function. The square root function is defined only for non-negative real numbers.

Step 2: Identify the Restrictions on the Variable

The given function $u(x) = \sqrt{-x} - 5$ has a restriction on the variable x, which is that x must be less than or equal to 0.

Step 3: Write the Domain in Interval Notation

The domain of the function $u(x) = \sqrt{-x} - 5$ can be written in interval notation as $(-\infty, 0]$ .

Conclusion

Final Answer

The final answer is $\boxed{(-\infty, 0]}$ .