Given g(x)= 1/3(x-2)^2-3 2<x<5 state the domain and image set g^-1 and find its rule
To determine the domain and image set of the function g(x) = 1/3(x-2)^2 - 3, we need to analyze the given information.
1. Domain:
The domain refers to the set of all possible values that x can take in the function. According to the given condition, the domain is stated as 2 < x < 5. This means that x must be greater than 2 and less than 5 for the function to be defined.
2. Image Set:
The image set represents the set of all possible values that g(x) can take. In this case, as the function is in the form of g(x) = 1/3(x-2)^2 - 3, it is evident that the graph is a parabola that opens upward. Since the coefficient of the squared term is positive (1/3), the minimum value occurs at the vertex of the parabola. Therefore, the image set starts from the vertex and extends to positive infinity. As a result, the image set of g(x) is (-∞, min(g(x)]).
Next, we need to find the rule for the inverse function, g^(-1)(x). To accomplish this, we can follow the following steps:
Step 1: Replace g(x) with y, so the equation becomes:
y = 1/3(x-2)^2 - 3
Step 2: Swap the x and y variables:
x = 1/3(y-2)^2 - 3
Step 3: Solve for y:
3(x + 3) = (y - 2)^2
√(3(x + 3)) = y - 2
y = √(3(x + 3)) + 2
Therefore, the rule for the inverse function g^(-1)(x) is:
g^(-1)(x) = √(3(x + 3)) + 2.