given no other restrictions, what are the domain and range of the following function?
f(x)=x^2-2x+2
the answer is D= all real numbers and R= {y|y>_1}
work out
To find the domain of the function, we need to determine all the possible x-values that the function can take. Since there are no restrictions given, the domain of the function is all real numbers. Therefore, D = all real numbers.
To find the range of the function, we need to determine all the possible y-values that the function can output. The range is the set of all possible values of y that can be obtained from plugging in different values of x into the function.
To find the range, we can start by finding the vertex of the quadratic function f(x) = x^2 - 2x + 2. The vertex form of a quadratic function is given by f(x) = a(x-h)^2 + k, where (h, k) is the vertex of the parabola.
In this case, a = 1, so the vertex form is f(x) = (x-1)^2 + 1. The vertex of this parabola is the point (h, k) = (1, 1).
Since the coefficient of the x^2 term is positive (+1), the parabola opens upwards. This means that the minimum value of the function occurs at the vertex. Therefore, the minimum value of f(x) is 1.
Since the minimum value of f(x) is 1, and the function can take any value greater than or equal to 1, the range of the function is R = {y | y ≥ 1}.
In summary, the domain of the function is D = all real numbers, and the range of the function is R = {y | y ≥ 1}.