## Hello! To determine the domain of the function y = âˆš(x^2 - 7x + 10), we need to find the values of x for which the expression inside the square root is non-negative.

Given that y = âˆš[(x-5)(x-2)], we can see that the expression inside the square root is (x-5)(x-2). In order for this to be non-negative, we need (x-5)(x-2) to be greater than or equal to zero.

To find the values of x that satisfy this inequality, we can consider the signs of each factor separately.

1. (x-5) â‰¥ 0:

To solve this inequality, we set x-5 greater than or equal to zero and solve for x:

x - 5 â‰¥ 0

x â‰¥ 5

2. (x-2) â‰¥ 0:

To solve this inequality, we set x-2 greater than or equal to zero and solve for x:

x - 2 â‰¥ 0

x â‰¥ 2

Now, let's combine the solutions for both factors. Since both factors must have the same sign for their product to be non-negative, we consider the intersection of the two solutions:

x â‰¥ 5 (from the first factor)

x â‰¥ 2 (from the second factor)

Taking the intersection of these two solutions, we find that the values of x that satisfy the inequality (x-5)(x-2) â‰¥ 0 are x â‰¥ 5.

Therefore, the domain of the function y = âˆš(x^2 - 7x + 10) expressed in interval notation is [5, âˆž).