the domain \( =(x-1)^{2}-9 \)

The expression \( (x-1)^{2}-9 \) represents a quadratic function, specifically a parabola that opens upwards. This parabola represents all real numbers because any real number \( x \) can be substituted into the equation, resulting in real outputs. The vertex of the parabola is at the point (1, -9), so the graph will stretch infinitely upwards from this lowest point. To find the range of this function, notice that \( (x-1)^2 \) is always greater than or equal to 0. Thus, the minimum value of the expression occurs when \( (x-1)^2 = 0 \), which leads to the lowest output of \(-9\). This means the function will take on values from \(-9\) to \(+\infty\). Isn’t it fascinating how quadratics reveal their secrets through their shape and position?