The function given in the image is $f(x) = \sqrt{x^2 - 9x + 14}$. To find the domain of this function, we need to ensure the expression under the square root is non-negative (i.e., $x^2 - 9x + 14 \geq 0$).

Let's solve the inequality $x^2 - 9x + 14 \geq 0$:

1. Factorize the quadratic expression: $x^2 - 9x + 14 = (x - 7)(x - 2)$

2. Use a sign chart or test values to determine where the product is non-negative:

   • For $x < 2$, both factors are negative, so their product is positive.
   • For $2 < x < 7$, one factor is negative and the other positive, so their product is negative.
   • For $x > 7$, both factors are positive, so their product is positive.

Therefore, the solution to the inequality $(x - 7)(x - 2) \geq 0$ is $x \in (-\infty, 2] \cup [7, \infty)$.

This interval represents the domain of $f(x)$.

Do you need any further details on this? Here are some related questions you might find interesting:

1. What would the domain of $f(x)$ be if the equation under the square root was $x^2 - 9x + 18$?
2. How would you graph the function $f(x) = \sqrt{x^2 - 9x + 14}$?
3. What are the critical points of the function $f(x)$, and how do they relate to its domain?
4. How does the domain change if you alter the coefficients of $x$ in the quadratic under the square root?
5. Could you solve for the range of the function $f(x) = \sqrt{x^2 - 9x + 14}$?

Tip: When determining the domain of functions involving square roots, always set the expression under the square root to be greater than or equal to zero and solve the resulting inequality. This approach is applicable for any square root function, ensuring that the argument remains real and defined.