Finding the Domain of Radical Function √3x² + 3 / 9

Video Solution

Solution Steps

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

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

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

00:11 A root must be for a positive number

00:16 Let's equate to 0 to find the solutions

00:24 Let's isolate X

00:32 Any number squared will always be positive, therefore there is no solution

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

Step-by-Step Solution

To determine the domain of the function $\frac{\sqrt{3x^2+3}}{9}$ , we need to ensure that the expression under the square root is non-negative:

$3x^2 + 3 \geq 0$

Simplifying this inequality, we can factor it:

Factor out the common term: $3(x^2 + 1) \geq 0$ .
Since $3$ is a positive constant, we focus on $x^2 + 1 \geq 0$ .
The term $x^2$ is always non-negative, hence $x^2 + 1$ is always positive for any real number $x$ , as the smallest value it can take, when $x = 0$ , is 1.

Thus, $x^2 + 1$ is never negative, making the expression under the square root always non-negative.

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

Emphasizing the conclusion: The entire domain of this function is all real numbers.