Final answer:
The inverse function g⁻¹(65) is found by solving the equation x² + 8x = 65. Completing the square and considering that x ≥ -4, the solution is x = 5, which means g⁻¹(65) = 5.
Step-by-step explanation:
To find g⁻¹(65), we first need to define the inverse function of g(x). Since g(x) = x² + 8x for x ≥ -4, we need to set g(x) equal to 65 and solve for x:
x² + 8x = 65
Now, we complete the square to make solving for x easier:
x² + 8x + (8/2)² = 65 + (8/2)²
x² + 8x + 16 = 65 + 16
(x + 4)² = 81
Next, we take the square root of both sides:
x + 4 = ±√81
x = -4 ± 9
Since x ≥ -4, we only consider the positive solution:
x = 9 - 4
x = 5
Therefore, g⁻¹(65) = 5, which is the value of x that makes g(x) equal to 65.