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

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

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

00:09 Therefore, from the denominator's perspective there is no domain restriction, let's check the numerator

00:12 A root must be for a positive number

00:17 Let's isolate X

00:32 The square of any number is always positive

00:36 Therefore there is no solution that makes the root negative

00:41 Thus the function exists for all X, and this is the solution to the question

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