Determine the domain and range of the quadratic function. (Enter your answers using interval notation.) f(x)=(x-1)^2+2 domain square range square

To determine the domain and range of the quadratic function \( f(x) = (x-1)^2 + 2 \), we need to analyze the function's behavior.### Domain:The domain of a quadratic function is all real numbers because there are no restrictions on the values that can take. Therefore, the domain is: ### Range:The range of the function depends on the vertex and the direction in which the parabola opens. 1. **Vertex**: The given function is in vertex form \( f(x) = a(x-h)^2 + k \), where \( (h, k) \) is the vertex of the parabola. For \( f(x) = (x-1)^2 + 2 \), the vertex is \( (1, 2) \).2. **Direction of Opening**: Since the coefficient of the squared term \((x-1)^2\) is positive (i.e., 1), the parabola opens upwards.3. **Minimum Value**: The minimum value of the function occurs at the vertex. Since the parabola opens upwards, the minimum value of \( f(x) \) is 2.Therefore, the range of the function is all values greater than or equal to 2. In interval notation, this is: So, the final answers are: