Given no other restrictions, what are the domain and range of the following function? f(x)=x2−2x+2
The given function is f(x) = x^2 - 2x + 2.
The domain of a function is the set of all possible input values (x-values). Since there are no restrictions given, the domain of this function is all real numbers, or (-∞, +∞).
To determine the range of the function, we can analyze the graph of the quadratic function. Since it is a parabolic function opening upwards (positive leading coefficient), the minimum point (vertex) of the parabola represents the lowest point on the graph.
Using the vertex formula: x = -b/2a, we find that x = -(-2)/(2*1) = 1. Therefore, the vertex is located at x = 1.
To find the y-coordinate of the vertex, we substitute x = 1 into the function:
f(1) = 1^2 - 2(1) + 2 = 1 - 2 + 2 = 1.
So, the vertex is (1, 1).
Since the parabola opens upwards, the range of the function is all real numbers greater than or equal to the y-coordinate of the vertex. Therefore, the range is [1, +∞).