Find the domain and range of f(x)=√x^2-7x+12
To find the domain of the function, we need to determine all the values of x for which the expression under the square root is non-negative. In other words, we need to solve the inequality:
x^2 - 7x + 12 ≥ 0
Factoring the left side, we get:
(x - 3)(x - 4) ≥ 0
The solutions to this inequality are x ≤ 3 and x ≥ 4. Therefore, the domain of the function is the interval (-∞, 3] ∪ [4, ∞).
To find the range of the function, we can graph it or use calculus. One way to graph the function is to complete the square:
f(x) = √(x - 3)(x - 4)
= √[(x - 3/2)^2 - 1/4]
This shows that the graph of the function is a shifted and stretched version of the square root function √x, with the vertex at (3/2, -1/2). Since the square root function has a range of [0, ∞), the range of the function f(x) is [-1/2, ∞).