Domain Of $f(x) = \sqrt{2x-4}$: Find The Origin!

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Hey guys! Today, we're diving into a fun little math problem: figuring out the domain of the function f(x) = √(2x-4). In simpler terms, we need to find all the possible x values that we can plug into this function without causing any mathematical mayhem. Let's break it down step by step!

Understanding the Domain

So, what exactly is a domain? The domain of a function is the set of all input values (x-values) for which the function produces a real number as an output. In other words, it's the set of all x-values that you can legally plug into the function. For square root functions, like the one we have here, there's one big rule we need to remember: we can't take the square root of a negative number (at least, not if we want a real number answer). This is super important because it restricts the values of x that are allowed in the domain. The expression inside the square root, called the radicand, must be greater than or equal to zero. If the radicand is negative, the function will produce an imaginary number, which isn't what we are looking for when determining the domain in the set of real numbers. Thus, the key is to make sure 2x-4 is not negative. So we can start solving for it.

Finding the Domain of f(x) = √(2x-4)

Alright, let's get our hands dirty and find the domain of f(x) = √(2x-4). As we discussed, the expression inside the square root (the radicand) must be greater than or equal to zero. This gives us the following inequality:

2x - 4 ≥ 0

Our goal is to isolate x and solve for it. To do this, we'll first add 4 to both sides of the inequality:

2x ≥ 4

Next, we'll divide both sides by 2:

x ≥ 2

This tells us that the domain of the function f(x) = √(2x-4) is all real numbers x that are greater than or equal to 2. Easy peasy!

Expressing the Domain

Now that we've found the domain, let's express it in different ways. There are a couple of common ways to represent the domain:

1. Interval Notation

In interval notation, we use brackets and parentheses to indicate the range of values for x. A square bracket [ ] means that the endpoint is included in the interval, while a parenthesis ( ) means that the endpoint is not included. Since our domain is x ≥ 2, we include 2 in the interval and extend to infinity. Infinity always gets a parenthesis because we can never actually reach infinity.

So, the interval notation for the domain is:

[2, ∞)

2. Set-Builder Notation

In set-builder notation, we use set notation to define the domain. We say that x is an element of the set of real numbers such that x is greater than or equal to 2. The set-builder notation looks like this:

{x ∈ ℝ | x ≥ 2}

This is read as "the set of all x in the set of real numbers such that x is greater than or equal to 2."

Why This Matters

You might be wondering, "Why do we even care about the domain of a function?" Well, the domain tells us where the function is actually defined and where it makes sense. In real-world applications, understanding the domain can help us avoid nonsensical results. For example, if our function represents the height of a plant over time, negative values of x (time) wouldn't make sense. Similarly, if the function involves physical constraints, such as the dimensions of an object, the domain will reflect those constraints. Imagine you're designing a rectangular garden. The function for the area of the garden might involve the length and width. Since the length and width can't be negative, the domain of the function would exclude negative values. In this case, knowing the domain helps you understand the limitations and practical implications of the function.

Common Mistakes to Avoid

When finding the domain of a function, it's easy to make a few common mistakes. Here are some to watch out for:

  1. Forgetting the Square Root Rule: The most common mistake is forgetting that the expression inside a square root must be greater than or equal to zero. Always double-check this condition.
  2. Incorrectly Solving the Inequality: Make sure you correctly solve the inequality. Remember to flip the inequality sign if you multiply or divide by a negative number.
  3. Using the Wrong Notation: Be careful when expressing the domain in interval or set-builder notation. Use brackets [ ] when the endpoint is included and parentheses ( ) when it's not. Also, make sure you understand the set-builder notation symbols.
  4. Ignoring Other Restrictions: Sometimes, functions have other restrictions besides square roots. For example, fractions can't have a zero denominator. Always consider all possible restrictions when finding the domain.

Example Problems

Let's try a couple more examples to practice finding the domain:

Example 1

Find the domain of g(x) = √(5 - x).

Solution:

We need to make sure that 5 - x ≥ 0. Adding x to both sides gives:

5 ≥ x

Which is the same as:

x ≤ 5

In interval notation, the domain is (-∞, 5]. In set-builder notation, the domain is {x ∈ ℝ | x ≤ 5}.

Example 2

Find the domain of h(x) = √(3x + 6).

Solution:

We need to make sure that 3x + 6 ≥ 0. Subtracting 6 from both sides gives:

3x ≥ -6

Dividing by 3 gives:

x ≥ -2

In interval notation, the domain is [-2, ∞). In set-builder notation, the domain is {x ∈ ℝ | x ≥ -2}.

Conclusion

Finding the domain of a function is a fundamental skill in mathematics. By understanding the restrictions on input values, we can ensure that our functions produce meaningful results. Remember to consider all possible restrictions, such as square roots, fractions, and real-world constraints. With practice, you'll become a pro at finding domains! Keep up the great work, and happy math-ing!

So, the correct answer from the options provided is:

  • interval [2, ∞)
  • {x ∈ R | x ≥ 2}