Final answer:
The range of the function f(x) = x² - 2x + 3 is [2, ∞)
Step-by-step explanation:
The range of a function represents the set of all possible output values. To find the range of the given function f(x) = x² - 2x + 3, we need to analyze the graph of the function and determine the minimum or maximum value it attains. In this case, the function is a quadratic function, and since the coefficient of the x² term is positive (1), the graph opens upward, indicating a minimum value.
To find the x-coordinate of the vertex, we can use the formula x = -b/2a, where a and b are the coefficients of the quadratic function. In this case, a = 1 and b = -2. Plugging these values into the formula, we get x = -(-2)/(2*1) = 1. Therefore, the vertex occurs at x = 1.
Substituting this value back into the function, we find the corresponding y-coordinate of the vertex: f(1) = 1² - 2(1) + 3 = 1 - 2 + 3 = 2. Hence, the vertex is located at (1, 2). Since the graph opens upward, this means that the minimum value of the function is 2.
Therefore, the range of the function f(x) = x² - 2x + 3 is all values greater than or equal to 2. In interval notation, we can write it as [2, ∞).