Domain Of F(x) = √(2x - 4): Find The Solution!

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Hey guys! Let's break down how to find the domain of the function f(x) = √(2x - 4). It might sound intimidating, but it's totally doable once you understand the concept. So, grab your thinking caps, and let's dive in!

Understanding the Domain

When we talk about the domain of a function, we're essentially asking: "What are all the possible x values that we can plug into this function and get a real number back as an answer?" Think of it like this: the domain is the set of all allowed inputs for our function machine. Not every number is invited to the party! In the context of real-valued functions, the domain consists of all real numbers for which the function produces a real number. This means we need to avoid situations that lead to undefined results, such as division by zero or taking the square root of a negative number.

For example, if we had a function like f(x) = 1/x, we'd have to exclude x = 0 from the domain because dividing by zero is a big no-no in mathematics. Similarly, for functions involving square roots, we need to make sure that the expression under the square root is non-negative (i.e., greater than or equal to zero), because the square root of a negative number is not a real number. Identifying these restrictions is key to accurately determining the domain of a function, ensuring that we only consider the x values that yield valid and meaningful outputs.

Dealing with Square Roots

The tricky part with our function, f(x) = √(2x - 4), is the square root. Remember, you can't take the square root of a negative number and get a real number answer. So, whatever is inside the square root has to be greater than or equal to zero. This gives us an inequality to solve.

Specifically, the expression inside the square root, which is 2x - 4, must be greater than or equal to zero. This ensures that we are only taking the square root of non-negative numbers, which will result in real number outputs. To find the valid range of x values, we set up the inequality 2x - 4 ≥ 0. Solving this inequality will give us the domain of the function, indicating all the x values for which the function is defined in the realm of real numbers. In practical terms, this means we are looking for all the x values that make the expression inside the square root either zero or a positive number. This is a fundamental concept when dealing with functions involving square roots, as it directly affects the set of possible inputs for the function.

Solving the Inequality

Okay, let's solve the inequality: 2x - 4 ≥ 0.

  1. Add 4 to both sides: This isolates the term with x on one side of the inequality, making it easier to solve for x. Adding 4 to both sides maintains the balance of the inequality while moving us closer to finding the solution. The new inequality becomes 2x ≥ 4.
  2. Divide both sides by 2: This isolates x completely, giving us the solution to the inequality. Dividing by 2 ensures that we find the exact range of x values that satisfy the original condition. Remember that when dividing or multiplying both sides of an inequality by a negative number, you must reverse the inequality sign. However, since we are dividing by a positive number (2), we don't need to worry about that. This gives us x ≥ 2.

What this means is that any x value that is greater than or equal to 2 will work in our original function. If we plug in a number less than 2, we'll end up taking the square root of a negative number, which is a no-go!

Expressing the Domain

So, how do we write this down in a fancy mathematical way?

There are a couple of common ways to express the domain:

  • Set-builder notation: This looks like {x ∈ R | x ≥ 2}. Let's break that down:
    • x ∈ R means "x is an element of the set of real numbers." In other words, x is a real number.
    • | means "such that."
    • x ≥ 2 means "x is greater than or equal to 2."
    • So, putting it all together, {x ∈ R | x ≥ 2} means "the set of all real numbers x such that x is greater than or equal to 2."
  • Interval notation: This looks like [2, ∞). Let's break this down too:
    • [ means "including the endpoint." So, 2 is included in the domain.
    • ) means "not including the endpoint." Infinity (∞) is never included because it's not a specific number; it's a concept.
    • So, [2, ∞) means "all numbers from 2 up to infinity, including 2."

The Answer

Looking back at the choices you provided, the correct answers are:

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

Why the Other Options are Wrong

Let's quickly go through why the other options are incorrect:

  • b. interval [-2, 2]: This includes numbers less than 2, which would result in taking the square root of a negative number.
  • c. {x ∈ R | x ≤ 2}: This includes numbers less than 2, which, as we've established, are not allowed.
  • d. interval (-∞, 2]: Again, this includes numbers less than 2.

Key Takeaways

  • Domain: The set of all possible input values (x values) for which a function produces a real number output.
  • Square Roots: The expression inside a square root must be greater than or equal to zero.
  • Solving Inequalities: Use basic algebraic operations to isolate the variable and find the range of valid values.
  • Notation: Be comfortable with both set-builder notation and interval notation for expressing the domain.

Practice Makes Perfect

The best way to master finding domains is to practice! Try finding the domains of these functions:

  • g(x) = √(x - 5)
  • h(x) = √(3x + 9)
  • k(x) = √(10 - 2x)

Remember to set the expression inside the square root greater than or equal to zero and solve for x. Good luck, and happy math-ing!

I hope this helps you understand how to find the domain of a function with a square root! Let me know if you have any other questions. Keep practicing, and you'll get the hang of it in no time!