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

Question

Look at the following function:

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

What is the domain of the function?

Video Solution

Solution Steps

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 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.

Answer

All real numbers

Related Subjects

Domain of a Function Indefinite integral Inputing Values into a Function

Question Types

Domain of a Function: Multiple domains