Finding The Domain Of A Square Root Function: A Step-by-Step Guide

Hey math enthusiasts! Let's dive into a common problem: finding the domain of a square root function. Specifically, we'll break down how to determine the correct inequality to use when you're given a function like $f(x)=\sqrt{\frac{1}{2} x-10}+3$. This is a fundamental concept in algebra, so understanding it is super important! The domain of a function is essentially the set of all possible input values (x-values) for which the function is defined. In simpler terms, it's all the x-values that won't break the function. When dealing with square root functions, there's a key rule we need to remember.

The Core Concept: Square Roots and Non-Negative Numbers

The most important thing to remember here is that you can't take the square root of a negative number (at least not within the real number system). That's where complex numbers come in, but we're not going there today! This fundamental principle is the cornerstone of finding the domain of square root functions. So, if we see a square root in a function, we automatically know that whatever is inside the square root (the radicand) must be greater than or equal to zero. This is because the square root function is only defined for non-negative numbers. Any negative value inside the square root would lead to an undefined result (in the real number system), and therefore, that x-value wouldn't be part of the domain. Think of it this way: the square root function acts like a gatekeeper, only allowing non-negative numbers to pass through. Understanding the domain means finding all the values that the gatekeeper will allow.

Now, let's look at our specific function: $f(x)=\sqrt{\frac{1}{2} x-10}+3$. The entire expression under the square root, is $\frac{1}{2} x-10$. According to our rule, this expression must be greater than or equal to zero. Why? Because that's the only way to ensure the function is defined. If this expression becomes negative, we're trying to take the square root of a negative number, which isn't allowed. The "+ 3" outside the square root doesn't affect the domain itself. It just shifts the graph vertically. It's the square root that dictates the allowable x-values. The addition of 3 means the whole graph moves up by 3 units on the y-axis, but the x-values that can go into the equation, the domain, remain unchanged. It's just like having a regular square root function and then adding a constant to it. The key thing to remember is the condition applied to the square root.

Therefore, we need to set up an inequality to represent this condition. This inequality will help us determine the possible x-values for which the function is defined. The goal is to find all the x-values that make the expression inside the square root greater than or equal to zero. This will give us the domain of the function. Let's explore how we arrive at that inequality in the next section.

Setting Up the Inequality: A Deep Dive

Alright, so we've established the fundamental rule: the expression inside the square root must be greater than or equal to zero. Now, let's translate that into an inequality for our function $f(x)=\sqrt\frac{1}{2} x-10}+3$. The expression inside the square root is $\frac{1}{2} x-10$. We want this expression to be greater than or equal to zero. So, the correct inequality is $\frac{1{2} x-10 \geq 0$. This inequality accurately reflects the condition for the square root to be defined. The other options provided in the initial question are incorrect because they don't capture the entire expression under the square root, leading to an incorrect domain. For example, $\frac{1}{2} x \geq 0$ would only consider part of the expression, and $\sqrt{\frac{1}{2} x} \geq 0$ is a slightly different function altogether. The correct choice is option C. This is because it directly focuses on the radicand (the expression under the square root), ensuring it's always non-negative. This is the condition needed for the square root function to produce a real number result, making it a valid point in the domain.

When we solve this inequality, we'll find the range of x-values that satisfy it. This range will be the domain of the function. By solving the inequality, we'll pinpoint all the acceptable input values (x-values) that produce a valid output for our function. It is important to note that the "+ 3" in the original function does not change our inequality. It only changes the vertical shift of the graph, which has nothing to do with the horizontal values, or domain. The domain is only about where the function is defined horizontally. The constant on the end does not influence this in any way, shape or form. The goal is to isolate 'x' on one side of the inequality. The solution to the inequality gives the set of all possible x-values for the domain, which will be all real numbers that will make the function true.

Solving the Inequality: Finding the Domain

Now that we know the correct inequality, $\frac{1}{2} x-10 \geq 0$, let's solve it to find the domain of the function $f(x)=\sqrt{\frac{1}{2} x-10}+3$. Solving this inequality involves isolating x. Here's how we do it step-by-step:

1. Add 10 to both sides: To get rid of the -10 on the left side, we add 10 to both sides of the inequality: $\frac{1}{2} x - 10 + 10 \geq 0 + 10$. This simplifies to $\frac{1}{2} x \geq 10$.

2. Multiply both sides by 2: To isolate x completely, we multiply both sides of the inequality by 2: $2 * \frac{1}{2} x \geq 10 * 2$. This simplifies to $x \geq 20$.

So, the solution to the inequality is $x \geq 20$. This means the domain of the function $f(x)=\sqrt{\frac{1}{2} x-10}+3$ consists of all x-values that are greater than or equal to 20. In interval notation, this is written as [20, \infty). This indicates that the function is defined for all real numbers from 20 (inclusive) to positive infinity. Any value less than 20 would result in a negative number inside the square root, which is not allowed in the real number system, making the function undefined for those values. This process is how we take the conditions on a square root function and use it to find the domain of the function, which is the set of valid x-values.

That's it! We've successfully found the domain by using the inequality. Remember, the key is understanding that the expression inside the square root must be greater than or equal to zero. Then, we solve the inequality to find the range of valid x-values.

Why the Other Options Are Incorrect

Let's quickly address why the other options provided in the initial question are incorrect, so you have a complete understanding.

• **A. $\sqrt\frac{1}{2} x} \geq 0$** This inequality considers only part of the original function's square root expression. It doesn't account for the "- 10". If we were dealing with the function $g(x) = \sqrt{\frac{1{2}x}$, then this inequality would be relevant. However, for our function, it's incomplete and will lead to an incorrect domain.

• B. $\frac{1}{2} x \geq 0$: This is similar to option A. It ignores the "- 10". It considers a portion of the radicand, but not the entire expression that's under the square root. Therefore, it would not find the right domain.

These two options will lead to a domain of $x \geq 0$, which is too broad and includes values that would make the original function undefined. The goal is to always make sure you are looking at the whole square root.

Understanding why the other options are wrong is crucial for solidifying your understanding. Always make sure to consider the entire expression under the square root when determining the domain. Don't be fooled by partial expressions; the whole expression must be considered to correctly determine the appropriate inequality and, therefore, the domain of the function.

Conclusion: Mastering the Domain of Square Root Functions

So, there you have it, guys! We've successfully navigated the process of finding the domain for a square root function. To recap:

• Recognize that the radicand (the expression under the square root) must be greater than or equal to zero.
• Set up the appropriate inequality based on the radicand.
• Solve the inequality to determine the range of x-values that make the function defined. This range is the domain.

This is a fundamental skill in algebra, and with practice, you'll be able to quickly determine the domain of any square root function. Keep practicing, and you'll become a domain expert in no time! Keep in mind that understanding the concept of a function's domain is important in many areas of mathematics. Now go forth and conquer those square roots!