Domain Of F(x) = √(3/(3+x)): Interval Notation Guide

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Hey guys! Today, we're diving into a fun math problem: finding the domain of the function f(x) = √(3/(3+x)). Don't worry, it's not as scary as it looks. We'll break it down step by step and express our answer in interval notation. So, grab your thinking caps, and let's get started!

Understanding the Domain

Before we jump into the specifics of this function, let's quickly recap what the domain actually means. In simple terms, the domain of a function is the set of all possible input values (x-values) that will produce a real number as an output (y-value). Think of it as the range of numbers you're allowed to plug into the function without causing any mathematical mayhem.

For this particular function, f(x) = √(3/(3+x)), we need to consider two key restrictions:

  1. Square Roots: We can't take the square root of a negative number and get a real number result. So, the expression inside the square root must be greater than or equal to zero.
  2. Division by Zero: We can't divide by zero. It's a big no-no in the math world. So, the denominator of any fraction must not be equal to zero.

Keeping these two restrictions in mind, let's tackle our function and find its domain.

Step-by-Step Solution

Okay, let's break down how to find the domain of f(x) = √(3/(3+x)) step by step. Remember, the goal is to figure out all the x values that we can plug into this function without causing any mathematical errors (like taking the square root of a negative number or dividing by zero).

1. Focus on the Square Root

The first thing we need to consider is the square root. We know that we can't take the square root of a negative number and get a real result. Therefore, the expression inside the square root must be greater than or equal to zero. In our case, the expression inside the square root is 3/(3+x). So, we have the inequality:

3/(3+x) ≥ 0

2. Analyze the Inequality

Now, let's analyze this inequality. We have a fraction, 3/(3+x), that needs to be greater than or equal to zero. The numerator, 3, is always positive. So, for the entire fraction to be positive (or zero), the denominator, (3+x), must also be positive. This gives us:

3 + x > 0

Note: We use a strict greater than (>) sign here because if 3 + x = 0, then we would be dividing by zero, which is not allowed.

3. Solve for x

Let's solve this inequality for x. Subtract 3 from both sides:

x > -3

This tells us that x must be greater than -3.

4. Consider Division by Zero

We've already taken care of the square root issue, but we also need to make sure we're not dividing by zero. The denominator of our fraction is (3+x). So, we need to make sure that:

3 + x ≠ 0

If we solve for x, we get:

x ≠ -3

This confirms what we found earlier when analyzing the inequality – x cannot be equal to -3.

5. Express the Domain in Interval Notation

Alright, we've figured out that x must be greater than -3, and x cannot be equal to -3. Now, let's express this in interval notation. Interval notation is a way of writing sets of numbers using intervals. We use parentheses () to indicate that an endpoint is not included and brackets [] to indicate that an endpoint is included.

In our case, x is greater than -3, but not equal to -3. This means we use a parenthesis at -3. Since x can be any number greater than -3, it goes all the way to positive infinity, which we also represent with a parenthesis (because infinity is not a specific number).

Therefore, the domain of the function f(x) = √(3/(3+x)) in interval notation is:

(-3, ∞)

Common Mistakes to Avoid

When finding the domain of functions, especially those involving square roots and fractions, it's easy to make a few common mistakes. Let's highlight some of these so you can steer clear of them.

Forgetting About Division by Zero

This is a big one! It's crucial to remember that you can't divide by zero. Always check the denominator of any fraction in your function and make sure it's not equal to zero. In our example, we had to ensure that (3 + x) ≠ 0.

Incorrectly Solving Inequalities

When dealing with inequalities, it's essential to solve them correctly. Remember the rules for inequalities, such as flipping the inequality sign when multiplying or dividing by a negative number. In our case, we had to solve 3/(3 + x) ≥ 0, which led us to x > -3.

Mixing Up Interval Notation

Interval notation can be a bit tricky at first. Remember that parentheses () mean the endpoint is not included, and brackets [] mean the endpoint is included. Also, always use parentheses with infinity (∞) and negative infinity (-∞) because they are not specific numbers.

Neglecting the Square Root Restriction

When a function involves a square root, remember that the expression inside the square root must be greater than or equal to zero. Forgetting this restriction can lead to an incorrect domain.

Not Double-Checking Your Answer

It's always a good idea to double-check your answer. Pick a few values within your calculated domain and plug them into the original function to see if they produce real results. Also, try a value outside your domain to confirm that it leads to an error.

By being aware of these common pitfalls, you'll be better equipped to find the domains of various functions accurately.

Practice Problems

To really nail down this concept, let's try a couple of practice problems. This will help you solidify your understanding of how to find the domain of functions with square roots and fractions. Remember to consider both the square root restriction (the expression inside must be greater than or equal to zero) and the division by zero restriction (the denominator cannot be zero).

Practice Problem 1

Find the domain of the function:

g(x) = √(5/(x - 2))

Practice Problem 2

Determine the domain of the function:

h(x) = √(4 - x) / (x + 1)

Work through these problems step-by-step, and don't hesitate to review the solution process we used for the original example. Pay close attention to setting up the correct inequalities and expressing your final answer in interval notation.

Hint: For Practice Problem 2, you'll need to consider both the numerator (the square root) and the denominator.

Conclusion

So there you have it! We've successfully found the domain of the function f(x) = √(3/(3+x)) and expressed it in interval notation as (-3, ∞). Remember the key steps: identify restrictions (square roots and division by zero), set up inequalities, solve for x, and write your answer in interval notation.

Finding the domain of a function might seem tricky at first, but with practice, you'll become a pro in no time. Keep exploring different functions and challenging yourself. You've got this!

If you have any questions or want to dive deeper into this topic, feel free to ask. Happy problem-solving, guys!