Domain Of F(x) = Sqrt(4-x^2): Find The Range!
Alright, let's dive into finding the domain of the function f(x) = √(4 - x²). This is a classic problem in mathematics, and understanding how to solve it will help you tackle similar problems with confidence. So, grab your thinking caps, and let's get started!
Understanding the Problem
Before we jump into calculations, let's make sure we understand what the question is asking. The domain of a function is essentially the set of all possible input values (x-values) that will produce a real number as an output. In other words, it's the set of x-values for which the function is defined. When dealing with square roots, we need to be particularly careful because the expression inside the square root (the radicand) must be non-negative. Why? Because the square root of a negative number is not a real number.
Why Non-Negative Radicand Matters
The key idea here is that we're working with real-valued functions. In the realm of real numbers, the square root of a negative number is undefined. For example, √(-1) is not a real number; it's an imaginary number, denoted as i. Since we're looking for the domain of f(x) within the real number system, we must ensure that the radicand (4 - x²) is always greater than or equal to zero.
Setting Up the Inequality
Now that we know the radicand must be non-negative, we can set up the following inequality:
4 - x² ≥ 0
This inequality tells us that the expression 4 - x² must be either positive or zero for the function f(x) to be defined. Solving this inequality will give us the range of x-values that satisfy this condition, which is the domain of our function.
Solving the Inequality
Okay, let's solve the inequality 4 - x² ≥ 0. There are a couple of ways to approach this, but we'll use a method that's both intuitive and reliable.
Rearranging the Inequality
First, let's rearrange the inequality to make it easier to work with. We can add x² to both sides:
4 ≥ x²
This is equivalent to:
x² ≤ 4
Taking the Square Root
Now, we need to take the square root of both sides. When we do this, we have to remember that taking the square root can result in both positive and negative values. So, we get:
-2 ≤ x ≤ 2
This means that x must be greater than or equal to -2 and less than or equal to 2.
Understanding the Result
What does this result tell us? It tells us that the values of x that make the function f(x) = √(4 - x²) defined are those between -2 and 2, inclusive. In other words, if we plug in any value of x within this range, we'll get a real number as an output. If we plug in any value outside this range, we'll get the square root of a negative number, which is not a real number.
Expressing the Domain
Now that we've found the range of x-values that satisfy the inequality, we can express the domain of the function f(x). There are a couple of ways to do this.
Interval Notation
In interval notation, the domain is written as:
[-2, 2]
This notation means that the domain includes all real numbers from -2 to 2, including -2 and 2 themselves. The square brackets indicate that the endpoints are included in the interval.
Set-Builder Notation
In set-builder notation, the domain is written as:
{x ∈ ℝ | -2 ≤ x ≤ 2}
This notation means "the set of all x in the set of real numbers such that x is greater than or equal to -2 and less than or equal to 2." It's a more formal way of expressing the same idea as interval notation.
Analyzing the Given Options
Now, let's take a look at the options provided in the question and see which one matches our result.
a. Interval [4, ∞): This is incorrect because it includes values greater than 2, which would result in a negative radicand. b. Interval [-2, 2]: This is correct! It matches the domain we found. c. Interval (-∞, -4] ∪ [4, ∞): This is incorrect because it includes values less than -2 and greater than 2, which would result in a negative radicand. d. x ∈ ℝ | -2 ≤ x ≤ 2}*: This is incorrect because it includes values greater than 2, which would result in a negative radicand.
Conclusion
So, the correct answers are:
b. Interval [-2, 2] d. {x ∈ ℝ | -2 ≤ x ≤ 2}
In summary, to find the domain of the function f(x) = √(4 - x²), we needed to ensure that the expression inside the square root was non-negative. By setting up and solving the inequality 4 - x² ≥ 0, we found that the domain is the interval [-2, 2], or in set-builder notation, {x ∈ ℝ | -2 ≤ x ≤ 2}. Understanding these concepts will help you tackle similar problems with greater ease and confidence. Keep practicing, and you'll become a domain-finding pro in no time!
Tips and Tricks
- Always remember that the radicand of a square root must be non-negative when working with real-valued functions.
- When solving inequalities involving square roots, be careful to consider both positive and negative roots.
- Practice expressing domains in both interval notation and set-builder notation to become comfortable with both.
- Use test values to verify your solution. Pick a value within the domain you found and plug it into the function to see if you get a real number. Pick a value outside the domain and see if you get the square root of a negative number.
Additional Exercises
To further practice finding domains, try these exercises:
- f(x) = √(9 - x²)
- g(x) = √(x² - 16)
- h(x) = √(25 - 4x²)
Work through these problems, and you'll solidify your understanding of finding domains of functions involving square roots.
Happy calculating, folks!