Find The Domain Of The Function And Write The Domain Using Interval Notation. F ( X ) = − 5 X − 4 X + 3 F(x) = \frac{-5x - 4}{x + 3} F ( X ) = X + 3 − 5 X − 4

Mar 13, 2025 by ADMIN 161 views

**Domain of a Rational Function: A Comprehensive Guide**

Introduction

In mathematics, the domain of a function is the set of all possible input values for which the function is defined. When dealing with rational functions, it's essential to determine the domain, as it can significantly impact the function's behavior and properties. In this article, we'll explore the concept of the domain of a rational function, focusing on the function $f(x) = \frac{-5x - 4}{x + 3}$ .

What is the Domain of a Rational Function?

The domain of a rational function is the set of all real numbers for which the function is defined. In other words, it's the set of all possible input values that can be plugged into the function without resulting in division by zero or any other undefined mathematical operation.

Key Concepts:

Before we dive into the specifics of the function $f(x) = \frac{-5x - 4}{x + 3}$ , let's review some essential concepts related to the domain of a rational function:

Denominator: The denominator of a rational function is the expression that appears in the denominator of the fraction. In the function $f(x) = \frac{-5x - 4}{x + 3}$ , the denominator is $x + 3$ .
Zero: A zero of a function is a value of the input variable that makes the function equal to zero. In the function $f(x) = \frac{-5x - 4}{x + 3}$ , the zero is $x = -3$ .
Undefined: A function is undefined at a point where the denominator is equal to zero, and the numerator is not equal to zero.

Finding the Domain of the Function

To find the domain of the function $f(x) = \frac{-5x - 4}{x + 3}$ , we need to determine the values of $x$ for which the function is defined. In other words, we need to find the values of $x$ that make the denominator $x + 3$ equal to zero.

Step 1: Set the Denominator Equal to Zero

To find the values of $x$ that make the denominator $x + 3$ equal to zero, we set the denominator equal to zero:

$x + 3 = 0$

Step 2: Solve for $x$

Now, we solve for $x$ :

$x = -3$

Step 3: Determine the Domain

Since the denominator is equal to zero when $x = -3$ , the function is undefined at this point. Therefore, the domain of the function $f(x) = \frac{-5x - 4}{x + 3}$ is all real numbers except $x = -3$ .

Writing the Domain Using Interval Notation

Interval notation is a way of representing a set of numbers using intervals. To write the domain of the function $f(x) = \frac{-5x - 4}{x + 3}$ using interval notation, we use the following notation:

$(-\infty, -3) \cup (-3, \infty)$

This notation indicates that the domain of the function is all real numbers except $x = -3$ .

Conclusion

In conclusion, the domain of a rational function is the set of all possible input values for which the function is defined. To find the domain of the function $f(x) = \frac{-5x - 4}{x + 3}$ , we set the denominator equal to zero, solve for $x$ , and determine the values of $x$ that make the function undefined. The domain of the function is all real numbers except $x = -3$ , which can be represented using interval notation as $(-\infty, -3) \cup (-3, \infty)$ .

Example Problems

Find the domain of the function $f(x) = \frac{2x - 1}{x - 2}$ .
Write the domain of the function $f(x) = \frac{x + 1}{x - 4}$ using interval notation.
Determine the domain of the function $f(x) = \frac{3x + 2}{x + 1}$ .

Solutions

To find the domain of the function $f(x) = \frac{2x - 1}{x - 2}$ , we set the denominator equal to zero and solve for $x$ :

$x - 2 = 0$

$x = 2$

The function is undefined at $x = 2$ , so the domain is all real numbers except $x = 2$ .

To write the domain of the function $f(x) = \frac{x + 1}{x - 4}$ using interval notation, we use the following notation:

$(-\infty, 4) \cup (4, \infty)$

This notation indicates that the domain of the function is all real numbers except $x = 4$ .

To determine the domain of the function $f(x) = \frac{3x + 2}{x + 1}$ , we set the denominator equal to zero and solve for $x$ :

$x + 1 = 0$

$x = -1$

The function is undefined at $x = -1$ , so the domain is all real numbers except $x = -1$ .

Final Thoughts

Q: What is the domain of a rational function?

A: The domain of a rational function is the set of all possible input values for which the function is defined. In other words, it's the set of all real numbers for which the function is not undefined.

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

A: To find the domain of a rational function, you need to determine the values of the input variable that make the denominator equal to zero. You can do this by setting the denominator equal to zero and solving for the input variable.

Q: What happens when the denominator of a rational function is equal to zero?

A: When the denominator of a rational function is equal to zero, the function is undefined at that point. This is because division by zero is undefined in mathematics.

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

A: Yes, a rational function can have a domain that includes all real numbers. This occurs when the denominator of the function is never equal to zero for any real number.

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

A: To write the domain of a rational function using interval notation, you need to identify the values of the input variable that make the function undefined. You can then use interval notation to represent the domain as a union of intervals.

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 for which the function is defined, while the range of a function is the set of all possible output values. In other words, the domain is the set of all possible inputs, while the range is the set of all possible outputs.

Q: Can a rational function have a domain that includes only a single point?

A: Yes, a rational function can have a domain that includes only a single point. This occurs when the denominator of the function is equal to zero at only one point.

Q: How do I determine the domain of a rational function with a variable in the denominator?

A: To determine the domain of a rational function with a variable in the denominator, you need to find the values of the variable that make the denominator equal to zero. You can then use these values to determine the domain of the function.

Q: Can a rational function have a domain that includes only a single interval?

A: Yes, a rational function can have a domain that includes only a single interval. This occurs when the denominator of the function is never equal to zero for any real number in the interval.

Q: How do I write the domain of a rational function with a variable in the denominator using interval notation?

A: To write the domain of a rational function with a variable in the denominator using interval notation, you need to identify the values of the variable that make the function undefined. You can then use interval notation to represent the domain as a union of intervals.

Q: What is the significance of the domain of a rational function?

A: The domain of a rational function is significant because it determines the set of all possible input values for which the function is defined. This is important because it can affect the behavior and properties of the function.

Q: Can a rational function have a domain that includes only a single point and a single interval?

A: Yes, a rational function can have a domain that includes only a single point and a single interval. This occurs when the denominator of the function is equal to zero at only one point and never equal to zero for any real number in the interval.

Q: How do I determine the domain of a rational function with a variable in the denominator and a constant in the numerator?

A: To determine the domain of a rational function with a variable in the denominator and a constant in the numerator, you need to find the values of the variable that make the denominator equal to zero. You can then use these values to determine the domain of the function.

Q: Can a rational function have a domain that includes only a single point and a single interval, and also include all real numbers?

A: Yes, a rational function can have a domain that includes only a single point and a single interval, and also include all real numbers. This occurs when the denominator of the function is equal to zero at only one point, never equal to zero for any real number in the interval, and never equal to zero for any real number outside the interval.

Q: How do I write the domain of a rational function with a variable in the denominator and a constant in the numerator using interval notation?

A: To write the domain of a rational function with a variable in the denominator and a constant in the numerator using interval notation, you need to identify the values of the variable that make the function undefined. You can then use interval notation to represent the domain as a union of intervals.

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

A: The domain and the range of a rational function are related in that the domain determines the set of all possible input values for which the function is defined, while the range determines the set of all possible output values. In other words, the domain is the set of all possible inputs, while the range is the set of all possible outputs.