The Domain of a Rational Function

Oct 13, 2025 by pynkocean.com 120 views

Domain of a Rational Function: Finding the Domain of $f(x) = \frac{x+7}{x-6}$

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 $x$ for which the function $f(x)$ is defined. In this article, we will discuss how to find the domain of a rational function, specifically the function $f(x) = \frac{x+7}{x-6}$ .

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. In other words, a rational function is a function that is the ratio of two polynomials.

The Domain of a Rational Function

The domain of a rational function is the set of all possible values of $x$ for which the function is defined. In other words, it is the set of all possible values of $x$ for which the denominator $q(x)$ is not equal to zero.

Finding the Domain of $f(x) = \frac{x+7}{x-6}$

To find the domain of the function $f(x) = \frac{x+7}{x-6}$ , we need to find the values of $x$ for which the denominator $x-6$ is not equal to zero.

Step 1: Set the Denominator Equal to Zero

To find the values of $x$ for which the denominator $x-6$ is not equal to zero, we need to set the denominator equal to zero and solve for $x$ . This will give us the values of $x$ for which the denominator is equal to zero.

$x - 6 = 0$

Step 2: Solve for $x$

To solve for $x$ , we need to isolate $x$ on one side of the equation.

$x = 6$

Step 3: Write the Domain in Interval Notation

The domain of the function $f(x) = \frac{x+7}{x-6}$ is the set of all possible values of $x$ for which the denominator $x-6$ is not equal to zero. In other words, it is the set of all possible values of $x$ except for $x=6$ .

The domain of the function $f(x) = \frac{x+7}{x-6}$ can be written in interval notation as:

$(-\infty, 6) \cup (6, \infty)$

In conclusion, the domain of the function $f(x) = \frac{x+7}{x-6}$ is the set of all possible values of $x$ except for $x=6$ . This can be written in interval notation as $(-\infty, 6) \cup (6, \infty)$ .

Why is the Domain Important?

The domain of a function is important because it tells us the set of all possible input values for which the function is defined. In other words, it tells us the set of all possible values of $x$ for which the function $f(x)$ is defined.

Real-World Applications

The concept of domain is important in many real-world applications, such as:

Physics: In physics, the domain of a function is used to describe the set of all possible values of a physical quantity, such as distance or time.
Engineering: In engineering, the domain of a function is used to describe the set of all possible values of a physical quantity, such as voltage or current.
Computer Science: In computer science, the domain of a function is used to describe the set of all possible input values for a function.

In conclusion, the domain of a function is an important concept in mathematics that tells us the set of all possible input values for which the function is defined. In this article, we discussed how to find the domain of a rational function, specifically the function $f(x) = \frac{x+7}{x-6}$ . We also discussed the importance of the domain in real-world applications and why it is an important concept to understand.
Domain of a Rational Function: Q&A

In our previous article, we discussed how to find the domain of a rational function, specifically the function $f(x) = \frac{x+7}{x-6}$ . In this article, we will answer some common questions related to 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 possible values of $x$ for which the function $f(x)$ is defined.

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

A: To find the domain of a rational function, you need to find the values of $x$ for which the denominator is not equal to zero. You can do this by setting the denominator equal to zero and solving for $x$ .

Q: What if the denominator is a polynomial of degree greater than 1?

A: If the denominator is a polynomial of degree greater than 1, you will need to factor the polynomial and set each factor equal to zero. This will give you multiple values of $x$ for which the denominator is equal to zero.

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

A: Yes, the domain of a rational function can be a single point. For example, if the denominator is a linear polynomial, the domain may be a single point.

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 polynomial, the domain may be an interval.

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

A: To write the domain of a rational function in interval notation, you need to use the following notation:

If the domain is a single point, use the notation $\{x\}$ .
If the domain is an interval, use the notation $[a, b]$ or $(a, b)$ .

Q: What if the domain of a rational function is a union of intervals?

A: If the domain of a rational function is a union of intervals, you can write it in interval notation using the following notation:

If the domain is a union of two intervals, use the notation $[a, b] \cup [c, d]$ or $(a, b) \cup (c, d)$ .
If the domain is a union of more than two intervals, use the notation $\cup_{i=1}^{n} [a_i, b_i]$ or $\cup_{i=1}^{n} (a_i, b_i)$ .

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

A: Yes, the domain of a rational function can be a union of single points. For example, if the denominator is a polynomial of degree greater than 1, the domain may be a union of single points.

In conclusion, the domain of a rational function is an important concept in mathematics that tells us the set of all possible input values for which the function is defined. In this article, we answered some common questions related to the domain of a rational function and provided examples of how to write the domain in interval notation.

Real-World Applications

The concept of domain is important in many real-world applications, such as:

Physics: In physics, the domain of a function is used to describe the set of all possible values of a physical quantity, such as distance or time.
Engineering: In engineering, the domain of a function is used to describe the set of all possible values of a physical quantity, such as voltage or current.
Computer Science: In computer science, the domain of a function is used to describe the set of all possible input values for a function.