Find The Domain Of The Function \[$ F(x) = |8-x| \$\].The Domain Is \[$\square\$\](Type Your Answer In Interval Notation.)

Feb 27, 2025 by ADMIN 123 views

**Finding the Domain of a Function: A Step-by-Step Guide**

Introduction

When dealing with functions, it's essential to understand the concept of the domain. The domain of a function is the set of all possible input values (x-values) for which the function is defined. In other words, it's the set of all possible x-values that the function can accept without resulting in an undefined or imaginary output. In this article, we'll focus on finding the domain of the function f(x) = |8-x|.

Understanding Absolute Value Functions

Before we dive into finding the domain of the function f(x) = |8-x|, let's take a closer look at absolute value functions. An absolute value function is a function that takes the absolute value of an expression as its output. The absolute value of a number is its distance from zero on the number line, without considering direction. In other words, it's the magnitude of the number.

The general form of an absolute value function is f(x) = |ax+b|, where a and b are constants. When a is positive, the graph of the function is a V-shaped graph that opens upwards. When a is negative, the graph of the function is a V-shaped graph that opens downwards.

Finding the Domain of f(x) = |8-x|

Now that we have a good understanding of absolute value functions, let's focus on finding the domain of the function f(x) = |8-x|. To find the domain of this function, we need to consider the possible input values (x-values) for which the function is defined.

The absolute value function f(x) = |8-x| is defined for all real numbers x. However, we need to consider the case when the expression inside the absolute value bars is equal to zero. This occurs when 8-x = 0, which gives us x = 8.

The Domain of f(x) = |8-x|

Based on our analysis, we can conclude that the domain of the function f(x) = |8-x| is all real numbers except x = 8. In interval notation, this can be written as (-∞, 8) ∪ (8, ∞).

Conclusion

In conclusion, finding the domain of a function is an essential step in understanding the behavior of the function. By considering the possible input values (x-values) for which the function is defined, we can determine the domain of the function. In this article, we focused on finding the domain of the function f(x) = |8-x| and concluded that the domain is all real numbers except x = 8.

Example Problems

Here are a few example problems to help you practice finding the domain of functions:

Find the domain of the function f(x) = |x-3|.
Find the domain of the function f(x) = |2x+1|.
Find the domain of the function f(x) = |x^2-4|.

Solutions

The domain of the function f(x) = |x-3| is all real numbers except x = 3.
The domain of the function f(x) = |2x+1| is all real numbers.
The domain of the function f(x) = |x^2-4| is all real numbers except x = 2 and x = -2.

Tips and Tricks

Here are a few tips and tricks to help you find the domain of functions:

Always consider the case when the expression inside the absolute value bars is equal to zero.
Use interval notation to write the domain of the function.
Make sure to include all possible input values (x-values) for which the function is defined.

Introduction

In our previous article, we discussed finding the domain of the function f(x) = |8-x|. We concluded that the domain of this function is all real numbers except x = 8. In this article, we'll answer some frequently asked questions about the domain of functions.

Q&A

Q: What is the domain of a function?

A: The domain of a function is the set of all possible input values (x-values) for which the function is defined.

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

A: To find the domain of a function, you need to consider the possible input values (x-values) for which the function is defined. You should also consider the case when the expression inside the absolute value bars is equal to zero.

Q: What is the difference between the domain and the range of a function?

A: The domain of a function is the set of all possible input values (x-values) for which the function is defined. The range of a function is the set of all possible output values (y-values) that the function can produce.

Q: Can a function have an empty domain?

A: Yes, a function can have an empty domain. This occurs when the function is not defined for any input value (x-value).

Q: Can a function have a domain that is all real numbers?

A: Yes, a function can have a domain that is all real numbers. This occurs when the function is defined for all possible input values (x-values).

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

A: To write the domain of a function in interval notation, you need to consider the possible input values (x-values) for which the function is defined. You should also use the following notation:

(-∞, a) to represent all real numbers less than a
(a, ∞) to represent all real numbers greater than a
[a, b] to represent all real numbers between a and b (inclusive)
(a, b) to represent all real numbers between a and b (exclusive)

Q: Can a function have a domain that is a union of intervals?

A: Yes, a function can have a domain that is a union of intervals. This occurs when the function is defined for multiple intervals of input values (x-values).

Q: How do I determine if a function is defined for a particular input value (x-value)?

A: To determine if a function is defined for a particular input value (x-value), you need to check if the function is defined at that point. You can do this by plugging in the input value (x-value) into the function and checking if the output is defined.