Domain & Range: 8x - 6y = -5 In Interval Notation

Hey guys! Today, we're diving deep into the fascinating world of relations and functions, specifically focusing on how to determine the domain and range of a linear equation. We'll be using interval notation, which is a super handy way to express sets of numbers. Our main example is the relation 8x - 6y = -5. So, buckle up, and let's get started!

Understanding Domain and Range

Before we jump into solving our specific problem, let's quickly recap what domain and range actually mean in the context of mathematical relations and functions. This is crucial for truly understanding what we're doing and not just memorizing steps.

• Domain: Think of the domain as all the possible input values for a relation. In simpler terms, it's the set of all x-values that you can plug into your equation. If you imagine a function as a machine, the domain is everything you're allowed to feed into it.
• Range: Now, the range is the set of all possible output values. It's what comes out of the machine after you've put something in. In terms of our equation, the range represents all the possible y-values that result from the x-values in our domain.

For linear equations like the one we're tackling today, things are usually pretty straightforward. However, when you get into more complex relations and functions (like those involving square roots, fractions, or logarithms), figuring out the domain and range can become a bit trickier. Certain x-values might lead to undefined results (like dividing by zero or taking the square root of a negative number), which means they wouldn't be included in the domain. Similarly, certain restrictions might affect the possible y-values in the range.

But for our linear equation, 8x - 6y = -5, we'll see that we don't have to worry about any of these complications. This equation represents a straight line, and straight lines (that aren't vertical) have a domain and range that include all real numbers. We'll explore why in the next section!

Analyzing the Relation: 8x - 6y = -5

Okay, let's get down to business and analyze our relation: 8x - 6y = -5. The first thing we need to recognize is that this is a linear equation. It's in the form of Ax + By = C, where A, B, and C are constants. Linear equations, when graphed, produce straight lines. This is a key piece of information because it significantly simplifies finding the domain and range.

Why does being a straight line help? Well, think about it visually. Imagine a line stretching across a coordinate plane. Unless it's a vertical line, it will extend infinitely in both the horizontal and vertical directions. This infinite extension is what dictates our domain and range.

To solidify this understanding, let's consider what would limit our domain and range. Could we plug in any value for x and get a real number for y? Absolutely! There's no operation here (like division or square roots) that would cause us problems for certain x-values. Similarly, if we solve for y, we'll see that there's no restriction on the values y can take.

Let's actually solve for y to see this more clearly:

1. Start with the equation: 8x - 6y = -5
2. Subtract 8x from both sides: -6y = -8x - 5
3. Divide both sides by -6: y = (4/3)x + (5/6)

Now we have the equation in slope-intercept form (y = mx + b). We can see that for any value of x we plug in, we'll get a real number for y. There are no restrictions! This means our domain is all real numbers.

Similarly, if we think about the possible y-values, we can see that the line will extend infinitely upwards and downwards. There's no horizontal asymptote or any other barrier to limit the range. Therefore, the range is also all real numbers.

Expressing Domain and Range in Interval Notation

Now that we've established that both the domain and range of our relation are all real numbers, we need to express this using interval notation. Interval notation is a concise way to represent sets of numbers using intervals and symbols.

The key symbols you need to know are:

• (, ) : Parentheses indicate that the endpoint is not included in the interval. This is used for open intervals and infinity.
• [, ] : Brackets indicate that the endpoint is included in the interval. This is used for closed intervals.
• ∞ : Infinity. This represents a value that extends without bound in the positive direction.
• -∞ : Negative infinity. This represents a value that extends without bound in the negative direction.

Since our domain and range are all real numbers, they extend infinitely in both the positive and negative directions. Therefore, we use infinity and negative infinity in our interval notation. And because infinity is not a specific number (we can't "reach" infinity), we always use parentheses with it.

So, the interval notation for all real numbers is (-∞, ∞).

Therefore, for the relation 8x - 6y = -5:

• Domain: (-∞, ∞)
• Range: (-∞, ∞)

Why Interval Notation Matters

You might be wondering, "Why bother with interval notation? Why not just say 'all real numbers'?" While saying "all real numbers" is perfectly understandable, interval notation provides a more precise and standardized way to express sets of numbers, especially when dealing with more complex functions and relations.

For example, imagine a function where the domain is all real numbers except for 2. In simple terms, you could say, "The domain is all real numbers except 2." But in interval notation, you can express this more clearly and concisely as (-∞, 2) ∪ (2, ∞). The ∪ symbol represents the union of two sets, meaning we're combining all numbers from negative infinity to 2 (excluding 2) with all numbers from 2 to infinity (again, excluding 2).

As you progress in your mathematical journey, you'll encounter more and more situations where interval notation becomes invaluable. It's a fundamental tool for expressing domains, ranges, solutions to inequalities, and much more.

Common Mistakes to Avoid

Before we wrap up, let's quickly touch on some common mistakes students make when determining domain and range, especially when using interval notation. Being aware of these pitfalls can help you avoid them!

1. Forgetting to consider restrictions: As we discussed earlier, not all functions have a domain and range of all real numbers. Be mindful of operations like division (where the denominator can't be zero) and square roots (where the radicand can't be negative). Always think about potential restrictions before declaring the domain and range.
2. Confusing parentheses and brackets: Remember that parentheses indicate exclusion, while brackets indicate inclusion. Using the wrong symbol can drastically change the meaning of your interval notation. For example, (2, 5) represents all numbers between 2 and 5, excluding 2 and 5, while [2, 5] represents all numbers between 2 and 5, including 2 and 5.
3. Incorrectly using infinity: Always use parentheses with infinity (∞) and negative infinity (-∞). Since infinity is not a specific number, it cannot be included in an interval.
4. Flipping the interval: Make sure you write your intervals in the correct order, from the smallest value to the largest value. (-∞, 5) is correct, while (5, -∞) is not.
5. Not visualizing the graph: If you're struggling to determine the domain and range, try sketching a quick graph of the relation or function. Visualizing the graph can often make the domain and range much clearer.

Conclusion

So, there you have it! We've successfully determined the domain and range of the relation 8x - 6y = -5 using interval notation. We found that both the domain and range are (-∞, ∞), meaning they include all real numbers. We also discussed the importance of understanding domain and range, the advantages of using interval notation, and some common mistakes to watch out for.

Remember, practice makes perfect! The more you work with different relations and functions, the more comfortable you'll become with finding their domains and ranges. Keep exploring, keep learning, and you'll be a domain and range pro in no time! Good luck, guys!