Domain Of Rational Function F(x) = 10x/(x-8)

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Hey guys! Today, we're diving into the fascinating world of rational functions and, more specifically, how to find their domains. If you've ever scratched your head wondering what the domain of a function like f(x) = 10x/(x-8) is, you're in the right place. Let’s break it down step by step, so you’ll be a pro in no time!

Understanding Rational Functions

First, let's get the basics down. A rational function is simply a function that can be expressed as a ratio of two polynomials. In other words, it looks like f(x) = P(x) / Q(x), where P(x) and Q(x) are polynomials. Our function, f(x) = 10x/(x-8), fits this description perfectly. Here, P(x) = 10x and Q(x) = x - 8. So far, so good, right?

Now, why do we even care about the domain? Well, the domain of a function is the set of all possible input values (x-values) for which the function is defined. Think of it as the range of numbers you're allowed to plug into the function without causing any mathematical mayhem. For most polynomial functions, the domain is all real numbers. However, rational functions come with a little twist.

The catch with rational functions is that we need to avoid values of x that make the denominator, Q(x), equal to zero. Why? Because division by zero is undefined in mathematics. It's like trying to split a pizza into zero slices – it just doesn't make sense! So, to find the domain of a rational function, we need to identify those pesky values of x that make the denominator zero and exclude them from the domain.

Finding the Domain of f(x) = 10x/(x-8)

Okay, let’s apply this to our specific function, f(x) = 10x/(x-8). Our mission is to find the values of x that make the denominator, x - 8, equal to zero. To do this, we set the denominator equal to zero and solve for x:

x - 8 = 0

Adding 8 to both sides, we get:

x = 8

Aha! So, x = 8 is the value that makes the denominator zero. This means that f(x) is undefined when x = 8. Therefore, we must exclude x = 8 from the domain of f(x).

In interval notation, the domain of f(x) is all real numbers except x = 8. We can express this as:

(-∞, 8) ∪ (8, ∞)

This notation means that the domain includes all numbers from negative infinity up to 8 (but not including 8), and all numbers from 8 (but not including 8) to positive infinity. Basically, every real number gets a thumbs-up except for 8.

In summary, to find the domain of the rational function f(x) = 10x/(x-8), we identified that the denominator cannot be zero. We solved for the value of x that makes the denominator zero (x = 8) and excluded it from the domain. The domain is therefore all real numbers except 8, which we wrote in interval notation as (-∞, 8) ∪ (8, ∞). Great job, you've nailed it!

General Steps to Find the Domain of a Rational Function

To make sure we’re crystal clear, let’s recap the general steps for finding the domain of any rational function. These steps will come in handy whenever you encounter a rational function, no matter how complex it looks.

  1. Identify the Rational Function: Make sure the function is in the form f(x) = P(x) / Q(x), where P(x) and Q(x) are polynomials.
  2. Set the Denominator Equal to Zero: Write the equation Q(x) = 0.
  3. Solve for x: Find all values of x that satisfy the equation Q(x) = 0. These are the values that make the denominator zero and must be excluded from the domain.
  4. Exclude the Values: The domain of the rational function is all real numbers except the values you found in step 3.
  5. Write the Domain in Interval Notation: Express the domain using intervals to show all the allowable values of x. For example, if x ≠ a, the domain is (-∞, a) ∪ (a, ∞).

Following these steps will help you confidently find the domain of any rational function you come across. Remember, the key is to avoid division by zero!

Examples and Practice

Let’s solidify our understanding with a few more examples. Practice makes perfect, and these examples will help you become even more comfortable with finding the domains of rational functions.

Example 1:

Find the domain of g(x) = (x + 3) / (x - 5).

  1. Identify: The function is rational with P(x) = x + 3 and Q(x) = x - 5.
  2. Set Denominator to Zero: x - 5 = 0
  3. Solve for x: x = 5
  4. Exclude the Value: x = 5 must be excluded.
  5. Interval Notation: The domain is (-∞, 5) ∪ (5, ∞).

Example 2:

Find the domain of h(x) = 2 / (x + 1).

  1. Identify: The function is rational with P(x) = 2 and Q(x) = x + 1.
  2. Set Denominator to Zero: x + 1 = 0
  3. Solve for x: x = -1
  4. Exclude the Value: x = -1 must be excluded.
  5. Interval Notation: The domain is (-∞, -1) ∪ (-1, ∞).

Example 3:

Find the domain of k(x) = (x - 4) / (x^2 - 9).

  1. Identify: The function is rational with P(x) = x - 4 and Q(x) = x^2 - 9.
  2. Set Denominator to Zero: x^2 - 9 = 0
  3. Solve for x: This is a difference of squares, so (x - 3)(x + 3) = 0. Thus, x = 3 or x = -3.
  4. Exclude the Values: x = 3 and x = -3 must be excluded.
  5. Interval Notation: The domain is (-∞, -3) ∪ (-3, 3) ∪ (3, ∞).

These examples should give you a solid foundation for finding the domains of various rational functions. Remember to always focus on the denominator and identify any values of x that would make it equal to zero. Keep practicing, and you’ll become a domain-finding master!

Why is Finding the Domain Important?

You might be wondering, “Why all this fuss about finding the domain?” Well, understanding the domain of a function is crucial for several reasons. It helps us make sense of the function’s behavior, avoid mathematical errors, and accurately interpret results.

  1. Avoiding Undefined Results: As we’ve emphasized, division by zero is a big no-no in mathematics. By finding the domain, we ensure that we only plug in values that produce valid results. This is especially important in real-world applications where undefined results can lead to incorrect conclusions or even serious problems.

  2. Understanding Function Behavior: The domain gives us insights into where the function is “well-behaved.” It tells us the range of input values for which the function produces meaningful output values. This knowledge is essential for graphing functions, analyzing their properties, and making predictions about their behavior.

  3. Real-World Applications: In many real-world scenarios, functions are used to model various phenomena. For example, a function might represent the population growth of a species, the trajectory of a projectile, or the cost of producing a certain number of items. In these cases, the domain often has a practical interpretation. For instance, if x represents the number of items produced, the domain might be restricted to non-negative integers because you can’t produce a negative or fractional number of items.

  4. Calculus and Advanced Math: In calculus and other advanced math courses, understanding the domain of a function is essential for many operations, such as finding limits, derivatives, and integrals. These operations rely on the function being well-defined over a certain interval, and the domain tells us exactly what that interval is.

In short, finding the domain is not just a mathematical exercise; it’s a fundamental skill that helps us understand, interpret, and apply functions in various contexts. So, keep practicing, and you’ll be well-equipped to tackle any function that comes your way!

Conclusion

Alright, guys, that wraps up our deep dive into finding the domain of rational functions. We've covered the basics, worked through examples, and discussed why it's so important to understand the domain. Remember, the key is to identify those pesky values that make the denominator zero and exclude them from the domain. With a little practice, you'll be finding domains like a pro!

Keep up the great work, and don't hesitate to revisit this guide whenever you need a refresher. Happy calculating!