To find the domain and range of the function f(x) = x^2 - 5x + 6, we will analyze its characteristics.
Domain:
The domain of a function is the set of all possible input values (x-values) for which the function is defined. In this case, f(x) is a polynomial function, and polynomials are defined for all real numbers. Hence, the domain of f(x) is all real numbers, or (-∞, ∞).
Range:
The range of a function is the set of all possible output values (y-values). To determine the range of f(x), we need to find the minimum/maximum value of the function.
First, let's consider the coefficient of the x^2 term, which is positive. This indicates that the parabola opens upwards, meaning there is no maximum value. Additionally, since there is no x-term coefficient, the function does not shift vertically.
Next, we look for the x-coordinate of the vertex. Using the formula x = -b/2a (where a is the coefficient of x^2, and b is the coefficient of x), we find x = -(-5) / (2 * 1) = 5/2.
To find the corresponding y-coordinate (the value of f(x) at the vertex), substitute this x-value back into the function: f(5/2) = (5/2)^2 - 5(5/2) + 6 = 1/4.
Therefore, the vertex of the parabola is (5/2, 1/4). Since the parabola opens upwards, the minimum value occurs at the vertex.
Hence, the range of f(x) is [1/4, ∞).