Determine the Domain of the Function: √(5x²+2) over 10

Q: Why don't I need to solve 5x² + 2 = 0?

Great question! You only solve the inequality when it can equal zero. Since $ x^2 \geq 0 $ for all real numbers, $ 5x^2 + 2 $ is always at least 2, never zero or negative.

Q: What if the constant was negative instead of +2?

If we had $ 5x^2 - 8 $, then you would need to solve $ 5x^2 - 8 \geq 0 $ to find domain restrictions. The positive constant +2 is what makes this always defined!

Q: Does the denominator 10 affect the domain?

No! Since 10 is a non-zero constant, it doesn't create any restrictions. Only expressions that could be zero or undefined (like square roots of negative numbers) affect the domain.

Q: How do I know when a square root expression is always positive?

Look for patterns like $ x^2 + \text{positive number} $ or $ (x-a)^2 + \text{positive number} $. Since squares are never negative, adding a positive constant keeps it positive!

Q: What's the difference between domain and range here?

The domain is all possible x-values (all real numbers). The range would be all possible y-values, which starts from $ \frac{\sqrt{2}}{10} $ and goes to infinity.

Domain Finding with Square Root Expressions

Look at the following function:

$\frac{\sqrt{5x^2+2}}{10}$

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

00:09 Let's see if this function has a domain. If it does, we'll find it together.

00:14 In the denominator, we avoid division by zero. So no domain issues there.

00:20 Next, let's check the numerator for any restrictions.

00:24 Remember, for a root, we need a positive number.

00:28 Let's focus on isolating X. Here, we're finding X values.

00:41 The square of any number is always positive, right?

00:45 So, there isn't a solution that makes the root negative.

00:50 This means the function exists for all values of X. And that's our answer!

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Look at the following function:

$\frac{\sqrt{5x^2+2}}{10}$

What is the domain of the function?

Step-by-step solution

To find the domain of the function $\frac{\sqrt{5x^2+2}}{10}$ , we need to ensure that the expression under the square root is non-negative.

Let's examine the inequality:

5x^2 + 2 \geq 0

This is always true since the expression $5x^2$ (a non-negative value for all real $x$ ) added to 2 will always be greater than or equal to zero. Consequently, $5x^2 + 2$ never takes a negative value, confirming the square root is always defined.

Therefore, the function $\frac{\sqrt{5x^2+2}}{10}$ is defined for all real numbers.

In conclusion, the domain of the function is all real numbers.

Final Answer

All real numbers

Key Points to Remember

Essential concepts to master this topic

Rule: Expression under square root must be non-negative
Technique: Check if $5x^2 + 2 \geq 0$ for all values
Check: Minimum value occurs when $x = 0$ : $5(0)^2 + 2 = 2 > 0$ ✓

Common Mistakes

Avoid these frequent errors

Incorrectly solving the inequality under the square root
Don't solve $5x^2 + 2 \geq 0$ by setting it equal to zero and finding roots = wrong restrictions! Since $x^2 \geq 0$ always, adding 2 makes it always positive. Always recognize when expressions are inherently non-negative.

Practice Quiz

Test your knowledge with interactive questions

Given the following function:

$\frac{5-x}{2-x}$

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

FAQ

Everything you need to know about this question

Why don't I need to solve 5x² + 2 = 0?

Great question! You only solve the inequality when it can equal zero. Since $x^2 \geq 0$ for all real numbers, $5x^2 + 2$ is always at least 2, never zero or negative.

What if the constant was negative instead of +2?

If we had $5x^2 - 8$ , then you would need to solve $5x^2 - 8 \geq 0$ to find domain restrictions. The positive constant +2 is what makes this always defined!

Does the denominator 10 affect the domain?

No! Since 10 is a non-zero constant, it doesn't create any restrictions. Only expressions that could be zero or undefined (like square roots of negative numbers) affect the domain.

How do I know when a square root expression is always positive?

Look for patterns like $x^2 + \text{positive number}$ or $(x-a)^2 + \text{positive number}$ . Since squares are never negative, adding a positive constant keeps it positive!

What's the difference between domain and range here?

The domain is all possible x-values (all real numbers). The range would be all possible y-values, which starts from $\frac{\sqrt{2}}{10}$ and goes to infinity.

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