Finding the Domain of a Rational Function

Oct 13, 2025 by windailys.com 117 views

Introduction

In mathematics, the domain of a function is the set of all possible input values for which the function is defined. Rational functions, in particular, have a domain that is influenced by the presence of zeros in the denominator. In this article, we will explore the concept of the domain of a rational function and provide a step-by-step guide on how to find the domain of the given function $$r(x)=\frac{x-3}{x^2-4x}$$ in interval notation.

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 is the set of all possible input values for which the function produces a real output value. The domain of a rational function is influenced by the presence of zeros in the denominator, as division by zero is undefined.

Finding the Domain of a Rational Function

To find the domain of a rational function, we need to identify the values of x that make the denominator equal to zero. These values are called the zeros of the denominator. Once we have identified the zeros of the denominator, we can use interval notation to express the domain of the function.

Step 1: Factor the Denominator

The first step in finding the domain of a rational function is to factor the denominator. In this case, the denominator is $$x^2-4x$$. We can factor this expression as follows:

x^2-4x = x(x-4)

Step 2: Identify the Zeros of the Denominator

Now that we have factored the denominator, we can identify the zeros of the denominator by setting each factor equal to zero and solving for x. In this case, we have two factors: x and (x-4). Setting each factor equal to zero, we get:

x = 0

(x-4) = 0

Solving for x, we get:

x = 0

x = 4

Step 3: Express the Domain in Interval Notation

Now that we have identified the zeros of the denominator, we can express the domain of the function in interval notation. The domain of the function is all real numbers except for the zeros of the denominator. In this case, the zeros of the denominator are x = 0 and x = 4. Therefore, the domain of the function is:

(-\infty, 0) \cup (0, 4) \cup (4, \infty)

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 a rational function, we need to identify the values of x that make the denominator equal to zero and express the domain in interval notation. In this article, we have provided a step-by-step guide on how to find the domain of the given function $$r(x)=\frac{x-3}{x^2-4x}$$ in interval notation.

Domain of the Given Function

The domain of the given function $$r(x)=\frac{x-3}{x^2-4x}$$ is:

(-\infty, 0) \cup (0, 4) \cup (4, \infty)

Final Answer

$The final answer is: $\boxed{(-\infty, 0) \cup (0, 4) \cup (4, \infty)}$<br/> **Domain of a Rational Function: A Comprehensive Guide**$

Q&A: Domain of a Rational Function

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 is the set of all real numbers for which the function produces a real output value.

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

A: To find the domain of a rational function, you need to identify the values of x that make the denominator equal to zero. These values are called the zeros of the denominator. Once you have identified the zeros of the denominator, you can use interval notation to express the domain of the function.

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 x-values, while the range is the set of all y-values.

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

A: Yes, a rational function can have a domain of all real numbers if the denominator is never equal to zero. For example, the function $$f(x) = \frac{1}{x}$$ has a domain of all real numbers except for x = 0.

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

A: To express the domain of a rational function in interval notation, you need to identify the zeros of the denominator and use interval notation to express the domain. For example, if the zeros of the denominator are x = a and x = b, the domain of the function can be expressed as:

(-\infty, a) \cup (a, b) \cup (b, \infty)

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

A: Yes, a rational function can have a domain that is a single interval. For example, the function $$f(x) = \frac{x-1}{x-2}$$ has a domain of all real numbers except for x = 2, which can be expressed as:

(-\infty, 2) \cup (2, \infty)

Q: How do I determine if a rational function is defined at a particular point?

A: To determine if a rational function is defined at a particular point, you need to check if the denominator is equal to zero at that point. If the denominator is not equal to zero, the function is defined at that point.

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

A: Yes, a rational function can have a domain that is a union of intervals. For example, the function $$f(x) = \frac{x-1}{x^2-4x}$$ has a domain of all real numbers except for x = 0 and x = 4, which can be expressed as:

(-\infty, 0) \cup (0, 4) \cup (4, \infty)

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 a rational function, you need to identify the values of x that make the denominator equal to zero and express the domain in interval notation. We hope this Q&A article has provided you with a comprehensive guide to the domain of a rational function.