Final answer:
To find g−1(11) for the function g(x) = x² + 10x given x ≥ −5, we set g(x) to 11 and solve the quadratic equation x² + 10x - 11 = 0. Using the quadratic formula, we determine that the valid solution is x = 1, as it falls within the domain of the function.
Step-by-step explanation:
To find the inverse function value g−1(11), we need to set g(x) = x² + 10x equal to 11, and solve for x. Hence, the equation is x² + 10x = 11.
Subtracting 11 from both sides, we get x² + 10x - 11 = 0. To solve this quadratic equation, we can either factor it, if it is factorable, or use the quadratic formula, which is x = (-b ± √(b² - 4ac))/(2a), where a = 1, b = 10, and c = -11.
The quadratic formula will give us the values of x that make the equation true:
- x = [-(10) ± √((10)² - 4(1)(-11))]/(2(1))
- x = [-10 ± √(100 + 44)]/2
- x = [-10 ± √144]/2
- x = [-10 ± 12]/2
Thus, we have two possible solutions for x:
- x = (2/2) = 1 (which is valid since x ≥ −5)
- x = (-22/2) = -11 (which is not valid since x must be ≥ −5)
The valid solution for the inverse function value g−1(11) is x = 1.