Determine the Domain of √(x²+2)/3

Q: Why doesn't x² + 2 have any restrictions like other square root problems?

Because x² is always non-negative (zero or positive), and adding 2 makes it even larger! The smallest possible value is when x = 0, giving us $ 0^2 + 2 = 2 $, which is still positive.

Q: What if the problem was √(x² - 2) instead?

Then you'd need $ x^2 - 2 \geq 0 $, which means $ x^2 \geq 2 $. This gives x ≥ √2 or x ≤ -√2, creating actual domain restrictions!

Q: Does the denominator 3 affect the domain?

No! Since 3 is a non-zero constant, it never causes division by zero. Only expressions with variables in the denominator can create domain restrictions.

Q: How can I quickly tell if a square root function has domain restrictions?

Look inside the square root: if you see subtraction (like x² - 5), there might be restrictions. If you see addition to x² (like x² + 2), the domain is usually all real numbers.

Q: What's the difference between domain and range for this function?

Domain is all possible x-values (input) = all real numbers. Range is all possible y-values (output). Since the smallest value under the square root is 2, the range is $ y \geq \frac{\sqrt{2}}{3} $.

Domain Analysis with Square Root Functions

Look at the following function:

$\frac{\sqrt{x^2+2}}{3}$

What is the domain of the function?

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Does the function have a domain? If so, what is it?

00:03 The domain of the denominator is to avoid division by 0

00:08 Therefore, for the denominator there is no domain restriction, let's check the numerator

00:12 A square root must be of a positive number

00:17 Any number squared is necessarily greater than 0

00:23 Therefore, any positive number plus 2 will be positive

00:27 Therefore, all X values are valid, and this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Look at the following function:

$\frac{\sqrt{x^2+2}}{3}$

What is the domain of the function?

Step-by-step solution

To solve this problem, we'll determine the domain of the function $f(x) = \frac{\sqrt{x^2+2}}{3}$ .

First, consider the expression inside the square root, $x^2 + 2$ . In order for the square root to be defined for real numbers, the expression $x^2 + 2$ must be non-negative.

Let's analyze $x^2 + 2$ :

For any real number $x$ , the expression $x^2$ is always non-negative.
Adding 2 to $x^2$ means $x^2 + 2$ is always greater than or equal to 2.
Thus, $x^2 + 2 \geq 2 > 0$ for all real numbers $x$ .

Since the value under the square root is always positive for all real numbers, the square root, and hence the function $f(x)$ , is defined for all real numbers.

Therefore, the function has no restrictions on its domain other than the real number system itself. There are no variables in the denominator that can make it zero, as it is the constant 3.

Thus, the domain of the function is all real numbers.

The correct answer choice is: All real numbers.

Final Answer

All real numbers

Key Points to Remember

Essential concepts to master this topic

Rule: For square roots, the expression inside must be non-negative
Technique: Check if x² + 2 ≥ 0 for all real x
Check: Verify minimum value: when x = 0, x² + 2 = 2 > 0 ✓

Common Mistakes

Avoid these frequent errors

Setting the expression under the square root equal to zero
Don't solve x² + 2 = 0 to find domain restrictions = impossible equation with no real solutions! This misses that x² + 2 is always positive. Always check if the expression can ever be negative, not just zero.

Practice Quiz

Test your knowledge with interactive questions

$22(\frac{2}{x}-1)=30$

What is the domain of the equation above?

FAQ

Everything you need to know about this question

Why doesn't x² + 2 have any restrictions like other square root problems?

Because x² is always non-negative (zero or positive), and adding 2 makes it even larger! The smallest possible value is when x = 0, giving us $0^2 + 2 = 2$ , which is still positive.

What if the problem was √(x² - 2) instead?

Then you'd need $x^2 - 2 \geq 0$ , which means $x^2 \geq 2$ . This gives x ≥ √2 or x ≤ -√2, creating actual domain restrictions!

Does the denominator 3 affect the domain?

No! Since 3 is a non-zero constant, it never causes division by zero. Only expressions with variables in the denominator can create domain restrictions.

How can I quickly tell if a square root function has domain restrictions?

Look inside the square root: if you see subtraction (like x² - 5), there might be restrictions. If you see addition to x² (like x² + 2), the domain is usually all real numbers.

What's the difference between domain and range for this function?

Domain is all possible x-values (input) = all real numbers. Range is all possible y-values (output). Since the smallest value under the square root is 2, the range is $y \geq \frac{\sqrt{2}}{3}$ .

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