Domain Of Rational Function: Set-Builder & Interval Notation

Hey guys! Let's dive into finding the domain of a rational function, specifically using set-builder and interval notations. This might sound a bit intimidating, but trust me, it's super manageable once you grasp the core concept. We'll be working with the function f(x) = (2x - 4) / (3x + 5). So, buckle up, and let's get started!

Understanding the Domain of Rational Functions

First off, what exactly is the domain of a function? Simply put, the domain is the set of all possible input values (often x values) that the function can accept without causing any mathematical errors. Think of it like this: if you have a machine (our function) that processes numbers, the domain is the list of numbers you can safely feed into it without breaking the machine. For rational functions, which are fractions with polynomials in the numerator and denominator, there's one major thing we need to watch out for: division by zero.

Division by zero is a big no-no in the math world. It's undefined, and it will make our function go haywire. So, to find the domain of a rational function, our primary goal is to identify any x values that would make the denominator equal to zero. These values must be excluded from the domain. In our case, we have the function f(x) = (2x - 4) / (3x + 5). The denominator is 3x + 5. To find the values of x that make the denominator zero, we need to solve the equation 3x + 5 = 0. Subtracting 5 from both sides gives us 3x = -5, and then dividing by 3, we get x = -5/3. This is the critical value that we need to exclude from our domain. All other real numbers are fair game!

Now, let's talk about why this matters. Imagine plugging x = -5/3 into our function. The denominator becomes zero, and we end up with a fraction where we're dividing by zero, which, as we've established, is a mathematical impossibility. Therefore, x = -5/3 cannot be part of the domain. This understanding is crucial for accurately expressing the domain using set-builder and interval notations.

(a) Set-Builder Notation

Okay, so we know that the domain includes all real numbers except for x = -5/3. How do we express this using set-builder notation? Set-builder notation is a concise way of defining a set by specifying the properties that its members must satisfy. It has a general form that looks like this: {x | condition}, which is read as "the set of all x such that a certain condition is true". The vertical bar "|" is read as "such that".

In our case, the condition is that x cannot be equal to -5/3. So, we can write the domain in set-builder notation as: {x | x ≠ -5/3}. Let's break this down: x represents all real numbers, the vertical bar means "such that", and x ≠ -5/3 is the condition that x cannot be equal to -5/3. This notation elegantly captures the idea that the domain consists of all real numbers except for the single value that makes the denominator zero.

To further illustrate this, consider a number line. We'd shade the entire number line to represent all real numbers, but we'd put an open circle at x = -5/3 to indicate that this point is excluded. Set-builder notation is a symbolic way of representing this visual idea. It's precise and leaves no room for ambiguity. It clearly states the condition that x must satisfy to be part of the domain. So, {x | x ≠ -5/3} is our final answer in set-builder notation. It tells us exactly which values are allowed and which are not. This is a powerful tool in mathematics, allowing us to define sets with clarity and precision.

(b) Interval Notation

Now, let's tackle interval notation. Interval notation is another way to represent sets of real numbers, particularly intervals. It uses parentheses and brackets to indicate whether the endpoints of an interval are included or excluded. Parentheses "( )" are used to indicate that an endpoint is excluded, while brackets "[ ]" are used to indicate that an endpoint is included. Infinity ∞ and negative infinity -∞ are always enclosed in parentheses because they are not actual numbers and cannot be included in a closed interval.

In our case, the domain includes all real numbers except x = -5/3. This means we have two intervals to consider: the interval to the left of -5/3 and the interval to the right of -5/3. The interval to the left of -5/3 includes all numbers less than -5/3, which can be written as (-∞, -5/3). Note the parenthesis around -5/3, indicating that -5/3 is not included. The interval to the right of -5/3 includes all numbers greater than -5/3, which can be written as (-5/3, ∞). Again, we use a parenthesis around -5/3 to exclude it.

To represent the entire domain, we need to combine these two intervals. We use the union symbol "∪" to indicate the union of two sets. So, the domain in interval notation is (-∞, -5/3) ∪ (-5/3, ∞). This notation tells us that the domain includes all numbers from negative infinity up to (but not including) -5/3, as well as all numbers from -5/3 (again, not included) to positive infinity.

Visually, we can think of this as the entire number line with a hole at -5/3. Interval notation provides a concise and clear way to express this. It avoids the need for inequalities and clearly shows the intervals where the function is defined. Therefore, (-∞, -5/3) ∪ (-5/3, ∞) is the final answer for the domain in interval notation. This method is particularly useful when dealing with more complex domains that might involve multiple intervals or exclusions.

Wrapping It Up

So, there you have it! We've successfully determined the domain of the rational function f(x) = (2x - 4) / (3x + 5) using both set-builder and interval notations. The key takeaway here is that understanding the restrictions imposed by rational functions, specifically the avoidance of division by zero, is crucial for accurately identifying and expressing the domain. We found that the domain is all real numbers except for x = -5/3. In set-builder notation, this is written as {x | x ≠ -5/3}, and in interval notation, it's (-∞, -5/3) ∪ (-5/3, ∞).

Remember, mastering these notations is essential for further mathematical studies, especially in calculus and analysis. They provide a precise and efficient way to communicate information about sets and intervals, which are fundamental concepts in mathematics. Keep practicing, and you'll become a pro at determining domains in no time! If you have any more questions, feel free to ask. Happy problem-solving, guys!