Questions: Determine the domain and range of the quadratic function. (Enter your answers using interval notation. f(x)=(x-6)^2+7 domain range

Solution Steps

To determine the domain and range of the quadratic function \( f(x) = (x-6)^2 + 7 \), we need to consider the properties of quadratic functions. The domain of any quadratic function is all real numbers because there are no restrictions on the values that \( x \) can take. The range is determined by the vertex of the parabola. Since the parabola opens upwards (as the coefficient of the squared term is positive), the minimum value of the function is at the vertex, which is \( y = 7 \). Therefore, the range is all real numbers greater than or equal to 7.

The domain of the quadratic function \( f(x) = (x-6)^2 + 7 \) is all real numbers. This is because there are no restrictions on the values that \( x \) can take in a quadratic function. Therefore, we express the domain as: \[ \text{Domain} = (-\infty, \infty) \]

To find the range, we analyze the vertex of the parabola represented by the function. The vertex form of the function is \( f(x) = (x-6)^2 + 7 \), where the vertex is at the point \( (6, 7) \). Since the parabola opens upwards (the coefficient of the squared term is positive), the minimum value of the function is at the vertex, which is \( y = 7 \). Thus, the range of the function is: \[ \text{Range} = [7, \infty) \]

Final Answer