Find the domain of f(x)=√x^2-7x+12
To find the domain of a function, we need to find all possible values of the input variable that make the function undefined or not meaningful.
In this case, we have a square root function with the expression inside the square root being x^2 - 7x + 12. For the square root to be defined, the expression inside must be greater than or equal to zero (since we can't take the square root of a negative number).
So we need to solve the inequality x^2 - 7x + 12 ≥ 0.
We can factor this expression as (x - 3)(x - 4) ≥ 0.
The critical values of x are where the expression changes sign, which occur at x = 3 and x = 4. So we need to consider three cases:
1. x < 3: Both factors are negative, so the product is positive.
2. 3 ≤ x ≤ 4: One factor is negative and one is nonnegative, so the product is negative.
3. x > 4: Both factors are nonnegative, so the product is positive.
Therefore, the solution to the inequality is x ∈ (-∞, 3] ∪ [4, ∞).
So the domain of the function f(x) = √(x^2 - 7x + 12) is (-∞, 3] ∪ [4, ∞).