Consider the function f(x)=−x2+6 on the domain [0,∞).
Use the range of f(x) to determine the domain of f−1(x).
What is the domain of f−1(x)?
Answers:
−∞<x≤6
−∞<f−1(x)≤6
6<x<∞
−6<f−1(x)<6
−6<x<6
6<f−1(x)<∞
the domain of f-1(x) is the range of f(x)
So, just find the range.
To determine the domain of the inverse function, f^(-1)(x), we can look at the range of the original function, f(x). The range of f(x) is the set of all possible values that f(x) can take.
In this case, the function f(x) = -x^2 + 6 is a downward opening parabola. The vertex of the parabola is at (0, 6). As x increases from 0 towards positive infinity, the value of f(x) decreases towards negative infinity.
So, we can say that the range of f(x) is (-∞, 6].
Now, to determine the domain of the inverse function, we reflect the range of the original function across the line y = x. This means that the values in the range become the corresponding values in the domain of the inverse function.
Therefore, the domain of f^(-1)(x) is (-∞, 6]. So, the answer is: -∞ < f^(-1)(x) ≤ 6.