Given no other restrictions what are the domain and range of the following function
f(x) = x^2 - 2x + 2
The domain of the function is all real numbers because there are no restrictions on what values of x can be plugged into the function.
To determine the range of the function, we can rewrite it in vertex form. The vertex form of a quadratic function is given by f(x) = a(x - h)^2 + k, where (h, k) represents the vertex of the parabola.
Completing the square on the given function, we have f(x) = (x^2 - 2x) + 2. To complete the square, we need to add and subtract (-2/2)^2 = 1 to the expression inside the parentheses: f(x) = (x^2 - 2x + 1 - 1) + 2 = (x^2 - 2x + 1) - 1 + 2 = (x - 1)^2 + 1.
Comparing this to the vertex form, we can conclude that the vertex of the parabola is (1, 1), and since the function is in standard form (y = ax^2 + bx + c), a = 1.
Since a = 1 is positive, the parabola opens upwards. Therefore, the range of the function is all real numbers greater than or equal to the minimum y-value of the vertex, which is 1.
In summary:
Domain: all real numbers
Range: all real numbers greater than or equal to 1