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.<br /><br />### Domain:<br />The domain of a quadratic function is all real numbers because there are no restrictions on the values that \( x \) can take. Therefore, the domain is:<br />\[ \text{Domain: } (-\infty, \infty) \]<br /><br />### Range:<br />The range of the function depends on the vertex and the direction in which the parabola opens. <br /><br />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 \( (1, 2) \).<br /><br />2. **Direction of Opening**: Since the coefficient of the squared term \((x-1)^2\) is positive (i.e., 1), the parabola opens upwards.<br /><br />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.<br /><br />Therefore, the range of the function is all values greater than or equal to 2. In interval notation, this is:<br />\[ \text{Range: } [2, \infty) \]<br /><br />So, the final answers are:<br />\[ \text{Domain: } (-\infty, \infty) \]<br />\[ \text{Range: } [2, \infty) \]