Question: Determine the domain and range of \( f(x) = 6(x + 5)^2 - 5 \). Enter the solution in interval…

Determine the domain and range of \( f(x) = 6(x + 5)^2 - 5 \). Enter the solution in interval notation.

• Domain:
• Range:

Solution

To determine the domain and range of the function \( f(x) = 6(x+5)^2 - 5 \), we proceed as follows: First, we determine the domain of the function. 1. The given function is a polynomial, and polynomials are defined for all real numbers. Therefore, the domain is: \[ (-\infty, \infty) \] Next, we determine the range of the function. 2. Recognize that \( f(x) = 6(x+5)^2 - 5 \) is a quadratic function in the form \( a(x-h)^2+k \). 3. The coefficient of the squared term, \( 6 \), is positive, indicating the parabola opens upwards. 4. The vertex of the parabola is at \(x = -5\), and the minimum value occurs at \(f(-5)\). 5. Substitute \(x = -5\) into the function to find the minimum value: \[ f(-5) = 6((-5)+5)^2 - 5 = 6(0)^2 - 5 = -5 \] 6. Since the parabola opens upwards, the range starts from the minimum value, \(-5\). Therefore, the range is: \[ [-5, \infty) \]