Find The Domain: A Rational Function Example

October 11, 2025 by TextBrain Team 45 views

Hey guys! Today, we're diving into a fun little math problem that involves finding the domain of a rational function. Don't worry; it's not as scary as it sounds! We'll break it down step by step so everyone can follow along. Our function is:

$f(x) = \frac{(x + 6)(x + 13)}{x^2 + 5x + 6}$

The domain of a function basically asks: what are all the possible values of 'x' that we can plug into the function without causing any mathematical mayhem? For rational functions (that is, fractions with polynomials), the main thing we need to watch out for is division by zero. Because, as we all know, dividing by zero is a big no-no in the math world! So, let's get started and figure out what values of 'x' would make the denominator of our function equal to zero.

Step 1: Factor the Denominator

The denominator of our function is $x^2 + 5x + 6$ . To find out when this equals zero, we need to factor it. Factoring is like reverse multiplication – we're trying to find two binomials that multiply together to give us our quadratic expression. In this case, we're looking for two numbers that add up to 5 and multiply to 6. Those numbers are 2 and 3. So, we can factor the denominator like this:

$x^2 + 5x + 6 = (x + 2)(x + 3)$

Alright, that wasn't too bad, was it? Now we have a factored denominator, which makes it much easier to see when it will equal zero.

Step 2: Find the Values That Make the Denominator Zero

Now that we have our factored denominator, $(x + 2)(x + 3)$ , we need to find the values of 'x' that make this expression equal to zero. This happens when either $(x + 2) = 0$ or $(x + 3) = 0$ . Let's solve each of these equations:

For $(x + 2) = 0$ , we subtract 2 from both sides to get $x = -2$ .
For $(x + 3) = 0$ , we subtract 3 from both sides to get $x = -3$ .

So, we've found that when $x = -2$ or $x = -3$ , the denominator of our function becomes zero. These are the values we need to exclude from the domain because they would cause division by zero, which is undefined.

Step 3: Determine the Domain

The domain of the function is all real numbers except for the values that make the denominator zero. We found that those values are $x = -2$ and $x = -3$ . Therefore, the domain of our function is all real numbers except -2 and -3. We can write this in set notation as:

Dom $f = \mathbb{R} - \{-2, -3\}$

This means the domain of f is the set of all real numbers excluding -2 and -3. In other words, you can plug in any number you want into the function, except for -2 and -3, and you'll get a valid result. If you try to plug in -2 or -3, you'll end up dividing by zero, which is not allowed. So, that's why we exclude those values from the domain. Understanding domains and ranges are very helpful when working with functions.

Step 4: Check the numerator

Let's consider the numerator of the function: $(x + 6)(x + 13)$ . The numerator equals zero when x=-6 or x=-13. These values are important for finding the x-intercepts of the function. However, they do not affect the domain of the function, as they only make the function equal to zero, which is perfectly acceptable. The domain is only restricted by values that make the denominator zero.

Conclusion

So, the correct answer is:

d. Dom $f = \mathbb{R} - \{-2, -3\}$

And there you have it! We've successfully found the domain of the rational function. Remember, the key is to identify any values of 'x' that would make the denominator zero and exclude them from the domain. With a little bit of factoring and solving, you can tackle any rational function domain problem that comes your way. Keep practicing, and you'll become a domain-finding pro in no time!

Rational functions are a fundamental part of algebra and calculus, and grasping their properties, especially the domain, is crucial for more advanced topics. The domain, as we've seen, tells us which 'x' values are permissible. Let's delve deeper into why this is so important and explore related concepts.

Why Domain Matters

The domain of a function is not just a theoretical concept; it has practical implications. When we define a function, we're essentially setting up a mathematical machine. We input a value (x), and the machine processes it to produce an output (f(x)). But like any machine, there are limits to what it can handle. The domain specifies those limits. For rational functions, the limit is division by zero. When we encounter a value that makes the denominator zero, the function becomes undefined at that point. This is why we exclude such values from the domain. Understanding domains helps us avoid nonsensical results and ensures our mathematical operations are valid.

Identifying Vertical Asymptotes

Knowing the domain of a rational function also helps us identify vertical asymptotes. A vertical asymptote is a vertical line that the graph of the function approaches but never touches. These occur at the 'x' values that make the denominator zero, i.e., the values excluded from the domain. For our function, $f(x) = \frac{(x + 6)(x + 13)}{x^2 + 5x + 6}$ , we found that the domain is all real numbers except -2 and -3. This means that the graph of the function has vertical asymptotes at $x = -2$ and $x = -3$ . Vertical asymptotes provide valuable information about the behavior of the function, especially as 'x' approaches these critical values.

Holes in Rational Functions

Sometimes, a rational function might have a hole instead of a vertical asymptote at a certain 'x' value. This happens when a factor in the denominator cancels out with a factor in the numerator. For example, consider the function:

$g(x) = \frac{(x - 2)(x + 1)}{(x - 2)}$

Here, the factor $(x - 2)$ appears in both the numerator and the denominator. We can cancel it out to simplify the function:

$g(x) = x + 1$ , for $x \neq 2$

However, we must remember that the original function was undefined at $x = 2$ . Even though the simplified function is defined at $x = 2$ , the original function has a hole at that point. A hole is a point where the function is not defined, but the graph appears to be continuous. To find the coordinates of the hole, we plug the 'x' value into the simplified function. In this case, $g(2) = 2 + 1 = 3$ , so the hole is at the point $(2, 3)$ .

The Importance of Factoring

Factoring is a crucial skill when working with rational functions. It allows us to identify the values that make the denominator zero, which are essential for determining the domain and finding vertical asymptotes. Factoring also helps us simplify rational functions by canceling out common factors, which can reveal holes in the graph. There are various techniques for factoring, including:

Greatest Common Factor (GCF): Finding the largest factor that divides all terms in the expression.
Difference of Squares: Factoring expressions of the form $a^2 - b^2$ as $(a + b)(a - b)$ .
Quadratic Factoring: Factoring quadratic expressions of the form $ax^2 + bx + c$ into two binomials.
Grouping: Factoring expressions with four or more terms by grouping them into pairs.

The more comfortable you are with factoring, the easier it will be to work with rational functions and understand their properties.

How to Determine the Domain of Functions

Determining the domain of a function is a fundamental concept in mathematics. The domain is the set of all possible input values (often 'x') for which the function is defined. In simpler terms, it's the set of 'x' values that you can plug into the function without causing any mathematical errors, such as division by zero or taking the square root of a negative number.

Types of Functions and Their Domains

Different types of functions have different rules for determining their domains. Here are some common types of functions and how to find their domains:

Polynomial Functions: Polynomial functions, such as linear, quadratic, and cubic functions, have a domain of all real numbers. This means you can plug in any real number for 'x', and the function will be defined.
Rational Functions: Rational functions are functions that are expressed as a fraction, where the numerator and denominator are both polynomials. The domain of a rational function is all real numbers except for the values that make the denominator equal to zero. To find the domain, set the denominator equal to zero and solve for 'x'. The values you find are the ones that you need to exclude from the domain.
Radical Functions: Radical functions are functions that involve taking a root, such as a square root or a cube root. For square root functions, the expression inside the square root must be greater than or equal to zero. To find the domain, set the expression inside the square root greater than or equal to zero and solve for 'x'. For cube root functions, the domain is all real numbers because you can take the cube root of any real number, whether it's positive, negative, or zero.
Logarithmic Functions: Logarithmic functions have a domain of all positive real numbers. The argument of the logarithm (the expression inside the logarithm) must be greater than zero. To find the domain, set the argument of the logarithm greater than zero and solve for 'x'.
Exponential Functions: Exponential functions have a domain of all real numbers. You can raise any real number to any power, so there are no restrictions on the domain.

Steps to Determine the Domain

Here are the general steps to determine the domain of a function:

Identify the type of function: Determine whether the function is a polynomial, rational, radical, logarithmic, or exponential function.
Identify any restrictions: Based on the type of function, identify any restrictions on the input values. For example, if the function is rational, identify the values that make the denominator zero. If the function is a square root, identify the values that make the expression inside the square root negative.
Solve for 'x': Solve for 'x' to find the values that you need to exclude from the domain or the values that satisfy the restrictions.
Write the domain: Write the domain as the set of all real numbers except for the values that you excluded, or as the set of values that satisfy the restrictions.

Examples

Here are some examples to illustrate how to determine the domain of different types of functions:

Example 1: Find the domain of the function $f(x) = 3x^2 + 5x - 2$ .

This is a polynomial function, so the domain is all real numbers.
Example 2: Find the domain of the function $f(x) = \frac{x + 1}{x - 2}$ .

This is a rational function, so we need to find the values that make the denominator zero. Set $x - 2 = 0$ and solve for 'x'. We get $x = 2$ , so the domain is all real numbers except for 2.
Example 3: Find the domain of the function $f(x) = \sqrt{x - 3}$ .

This is a square root function, so we need to find the values that make the expression inside the square root greater than or equal to zero. Set $x - 3 \geq 0$ and solve for 'x'. We get $x \geq 3$ , so the domain is all real numbers greater than or equal to 3.
Example 4: Find the domain of the function $f(x) = \log(x + 4)$ .

This is a logarithmic function, so we need to find the values that make the argument of the logarithm greater than zero. Set $x + 4 > 0$ and solve for 'x'. We get $x > -4$ , so the domain is all real numbers greater than -4.

By following these steps and understanding the restrictions for different types of functions, you can confidently determine the domain of any function you encounter. Remember to always check for division by zero, square roots of negative numbers, and logarithms of non-positive numbers. These are the most common pitfalls when finding domains.