Domain Of F(x) = 3√(x-2): A Step-by-Step Explanation
Hey there, math enthusiasts! Today, we're diving deep into the fascinating world of functions, specifically focusing on how to determine the domain of a function. Our spotlight function is $f(x) = 3\sqrt{(x-2)}$. Understanding the domain is crucial because it tells us the set of all possible input values (x-values) for which the function will produce a real number output. So, let's roll up our sleeves and get started!
What is the Domain of a Function?
Before we tackle our specific function, let's quickly recap what the domain actually means. In simple terms, the domain of a function is the set of all x-values that you can plug into the function without causing any mathematical mayhem. Think of it like this: the function is a machine, and the domain is the list of ingredients that the machine can process. If you try to feed it something it can't handle, the machine might break down (or, in math terms, give you an undefined result).
For instance, some common culprits that can restrict a function's domain are:
- Division by zero: We can't divide any number by zero – it's a big no-no in the math world. So, if a function has a denominator, we need to make sure that the denominator never equals zero for any x-value in the domain.
- Square roots of negative numbers: In the realm of real numbers, we can't take the square root of a negative number. This is because there's no real number that, when multiplied by itself, gives you a negative result. (This changes when we venture into the world of complex numbers, but that's a story for another day!). So, if a function involves a square root (or any even root), we need to ensure that the expression under the root is non-negative (i.e., greater than or equal to zero).
- Logarithms of non-positive numbers: Logarithms are only defined for positive numbers. You can't take the logarithm of zero or a negative number. So, if a function includes a logarithm, we need to make sure that the argument of the logarithm is strictly positive.
Knowing these restrictions is the first step in finding the domain of any function. Now, let's apply this knowledge to our function, $f(x) = 3\sqrt{(x-2)}$.
Finding the Domain of $f(x) = 3\sqrt{(x-2)}$
Alright, let's get our hands dirty and figure out the domain of our function. Looking at $f(x) = 3\sqrt{(x-2)}$, we can immediately spot a potential issue: the square root. As we discussed earlier, we can't take the square root of a negative number and still get a real number result. This means that the expression inside the square root, which is (x - 2), must be greater than or equal to zero.
So, here's the key inequality we need to solve:
To solve for x, we simply add 2 to both sides of the inequality:
This inequality tells us that the x-values that are allowed in the domain are all the numbers greater than or equal to 2. Any x-value less than 2 would make the expression inside the square root negative, leading to an undefined result in the real number system. Guys, remember this is super important for understanding functions!
Therefore, the domain of the function $f(x) = 3\sqrt{(x-2)}$ is all real numbers x such that x is greater than or equal to 2. We can express this domain in several ways:
- Inequality notation: $x \geq 2$
- Interval notation: $[2, \infty)$
- Set-builder notation: $\lbrace x \in \mathbb{R} \mid x \geq 2 \rbrace$
Let's break down the interval notation, as it's commonly used and might be new to some of you. The square bracket '[' indicates that the endpoint 2 is included in the domain (because x can be equal to 2), and the parenthesis ')' indicates that infinity is not included (because infinity isn't a specific number, but rather a concept of unboundedness). The set-builder notation is a more formal way of saying the same thing:
