How to Find the Domain of a Rational Function

Oct 13, 2025 by miodeya.com 84 views

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, such as the one given in this problem, $f(x) = \frac{6x}{x-7}$ , we need to be careful about the values of $x$ that make the denominator equal to zero. In this article, we will explore how to find the domain of a rational function and apply this knowledge to the given function.

What is a Rational Function?

A rational function is a function that can be written in the form $\frac{p(x)}{q(x)}$ , where $p(x)$ and $q(x)$ are polynomials and $q(x)$ is not equal to zero. Rational functions are a type of algebraic function that can be used to model a wide range of real-world phenomena.

Why is the Domain of a Rational Function Important?

The domain of a rational function is important because it tells us the set of all possible input values for which the function is defined. If we try to evaluate a rational function at a value outside of its domain, we will get an undefined result, often represented as "not a number" or "NaN". Therefore, it is essential to find the domain of a rational function before trying to evaluate it.

How to Find 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. This is because division by zero is undefined in mathematics. We can do this by setting the denominator equal to zero and solving for $x$ .

Step 1: Set the Denominator Equal to Zero

Let's set the denominator of the given function, $x-7$ , equal to zero and solve for $x$ .

x-7=0

Step 2: Solve for $x$

Now, let's solve for $x$ by adding 7 to both sides of the equation.

x=7

Step 3: Write the Domain

The domain of the function is all real numbers except the value of $x$ that makes the denominator equal to zero. In this case, the value of $x$ is 7.

Conclusion

In conclusion, the domain of the function $f(x) = \frac{6x}{x-7}$ is all real numbers except 7. This means that the function is defined for all real numbers except 7.

Answer

The correct answer is:

B. All real numbers except 0 and 7

Final Thoughts

Introduction

In our previous article, we explored how to find the domain of a rational function. We saw that the domain of a rational function is the set of all possible input values for which the function is defined. In this article, we will answer some frequently asked questions about finding the 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 except the values that make the denominator equal to zero.

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. This is because division by zero is undefined in mathematics. You can do this by setting the denominator equal to zero and solving for $x$ .

Q: What if the denominator is a quadratic expression?

A: If the denominator is a quadratic expression, you will need to factor it and set each factor equal to zero. This will give you multiple values of $x$ that make the denominator equal to zero. You will need to exclude these values from the domain.

Q: Can the domain of a rational function be a single value?

A: Yes, the domain of a rational function can be a single value. For example, if the denominator is a linear expression, such as $x-7$ , the domain will be all real numbers except 7.

Q: Can the domain of a rational function be an interval?

A: Yes, the domain of a rational function can be an interval. For example, if the denominator is a quadratic expression, such as $x^2-4$ , the domain will be all real numbers except $x=-2$ and $x=2$ .

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

A: To write the domain of a rational function, you need to use interval notation. For example, if the domain is all real numbers except 7, you would write it as $(-\infty, 7) \cup (7, \infty)$ .

Q: Can the domain of a rational function be a union of intervals?

A: Yes, the domain of a rational function can be a union of intervals. For example, if the denominator is a quadratic expression, such as $x^2-4$ , the domain will be $(-\infty, -2) \cup (-2, 2) \cup (2, \infty)$ .

Q: How do I determine if a rational function is defined at a particular value of $x$ ?

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

Conclusion

In conclusion, finding the domain of a rational function is an essential step in understanding the behavior of the function. By identifying the values of $x$ that make the denominator equal to zero, we can determine the set of all possible input values for which the function is defined. We hope that this Q&A article has helped to clarify any questions you may have had about finding the domain of a rational function.

Common Mistakes to Avoid

Not setting the denominator equal to zero when finding the domain of a rational function.
Not solving for $x$ when setting the denominator equal to zero.
Not excluding the values of $x$ that make the denominator equal to zero from the domain.
Not using interval notation to write the domain of a rational function.

Final Thoughts

Finding the domain of a rational function is a crucial step in understanding the behavior of the function. By following the steps outlined in this article, you can ensure that you are finding the correct domain for a rational function. Remember to always set the denominator equal to zero, solve for $x$ , and exclude the values of $x$ that make the denominator equal to zero from the domain.