Domain Of (x-3)/(2x-8): A Quick Guide

by Felix Dubois 38 views

Hey guys! Ever wondered about the domain of rational expressions? It might sound intimidating, but trust me, it's not as scary as it seems. Think of it like this: a rational expression is just a fraction where the numerator and denominator are polynomials. And just like regular fractions, there are certain values that can cause problems – specifically, values that make the denominator zero. Let's dive in and break down how to find the domain of the rational expression (x-3)/(2x-8).

What is a Rational Expression?

Before we jump into the specifics, let's make sure we're all on the same page. A rational expression is essentially a fraction where both the numerator and the denominator are polynomials. Polynomials, remember, are expressions involving variables and coefficients, combined using addition, subtraction, and non-negative integer exponents (think things like x^2 + 3x - 5 or simply 7x + 2). So, a rational expression might look like (x^2 + 1) / (x - 3) or (5x) / (x^2 + 2x + 1). The key thing to remember is that the denominator cannot be equal to zero.

Why can't the denominator be zero? Well, division by zero is undefined in mathematics. It's a big no-no! Trying to divide by zero leads to all sorts of mathematical inconsistencies and paradoxes. So, when we're dealing with rational expressions, we need to be extra careful to identify any values of the variable that would make the denominator zero. These values are excluded from the domain of the expression.

The domain, in simple terms, is the set of all possible input values (usually represented by the variable 'x') for which the expression is defined and produces a real number output. In the context of rational expressions, the domain consists of all real numbers except those that make the denominator zero. Finding these excluded values is crucial for understanding the behavior of the rational expression and for performing operations like simplifying, adding, subtracting, multiplying, and dividing rational expressions.

Understanding the domain also becomes vital when you start graphing rational functions. The excluded values often correspond to vertical asymptotes on the graph, which are vertical lines that the graph approaches but never actually touches. These asymptotes play a significant role in shaping the overall appearance of the graph. So, grasping the concept of the domain is not just about avoiding division by zero; it's about gaining a deeper understanding of the function's behavior and its graphical representation.

Identifying the Problem: Division by Zero

The core issue when finding the domain of a rational expression is division by zero. We need to figure out what values of 'x' would make the denominator of our expression, 2x - 8, equal to zero. This is because, as we discussed, division by zero is undefined in the mathematical world. It's like trying to split a pizza among zero people – it just doesn't make sense!

To find these problematic values, we set the denominator equal to zero and solve for 'x'. This will give us the values that we need to exclude from our domain. In our case, the denominator is 2x - 8. So, we set up the equation 2x - 8 = 0. Now, it's just a matter of solving this simple algebraic equation.

Adding 8 to both sides of the equation, we get 2x = 8. Then, dividing both sides by 2, we find that x = 4. This is a crucial piece of information! It tells us that if we plug in x = 4 into the denominator (2x - 8), we get 2(4) - 8 = 8 - 8 = 0. Bingo! We've found the value that makes the denominator zero.

This means that x = 4 is not allowed in the domain of our rational expression. If we were to try and evaluate the expression (x - 3) / (2x - 8) at x = 4, we would end up dividing by zero, which is undefined. So, 4 is a value we need to exclude. But what about other values? Could there be more values that make the denominator zero? In this case, since the denominator is a linear expression (2x - 8), there's only one value that makes it zero. If the denominator were a quadratic or a higher-degree polynomial, there might be multiple values to exclude. But for our expression, x = 4 is the only culprit.

Understanding this step is paramount. It's the foundation for finding the domain of any rational expression. By identifying the values that make the denominator zero, we're essentially identifying the forbidden zones – the values that will break our mathematical rules and lead to undefined results. Once we've pinpointed these forbidden zones, we can confidently define the domain of the expression.

Solving for the Excluded Value

Alright, we've established that division by zero is the enemy, and we know we need to find the values of 'x' that make the denominator 2x - 8 equal to zero. Now, let's get our hands dirty and actually solve the equation. As we mentioned earlier, we set up the equation 2x - 8 = 0. This is a simple linear equation, and we can solve it using basic algebraic techniques.

Our goal is to isolate 'x' on one side of the equation. The first step is to get rid of the -8. To do this, we add 8 to both sides of the equation. Remember, whatever we do to one side of the equation, we must do to the other to maintain the balance. So, we have:

2x - 8 + 8 = 0 + 8

This simplifies to:

2x = 8

Now, we have 2x equal to 8. To isolate 'x', we need to get rid of the 2 that's multiplying it. We do this by dividing both sides of the equation by 2:

2x / 2 = 8 / 2

This gives us:

x = 4

And there you have it! We've solved the equation and found that x = 4 is the value that makes the denominator zero. This confirms what we suspected – that 4 is the value we need to exclude from the domain. Solving for the excluded value might seem straightforward in this case, but it's a crucial skill that applies to more complex rational expressions as well. If the denominator were a quadratic (like x^2 - 5x + 6), you would need to use factoring or the quadratic formula to find the values that make it zero.

But for our expression, (x - 3) / (2x - 8), the solution is clear: x = 4. This value is the key to understanding the domain of this rational expression. It's the one number that's off-limits, the one value that would lead to an undefined result. So, how do we express this in terms of the domain? That's the next step!

Defining the Domain: All Real Numbers Except…

Now that we've pinpointed the value that makes the denominator zero (x = 4), we can finally define the domain of the rational expression (x - 3) / (2x - 8). Remember, the domain is the set of all possible input values for 'x' that don't lead to undefined results. In this case, that means all real numbers except for 4.

There are a couple of ways we can express this mathematically. One way is using set notation. In set notation, we write the domain as:

{x | x ∈ ℝ, x ≠ 4}

Let's break this down:

  • '{x | ...}' means