Final answer:
A piecewise function satisfying the domain (-3, 2) U (2,∞) and range (-1,∞) could be f(x) = { x² - 1 for -3 < x < 2, and x - 1 for x > 2 }, ensuring that for the entire range of x, the values of f(x) are greater than or equal to -1.
Step-by-step explanation:
To create a piecewise function with the given domain and range, we need to define separate expressions for different parts of the domain. The domain is split into two intervals: (-3, 2) and (2, ∞), with a discontinuity at x = 2. The range starting from -1 and extending to ∞ suggests that the output of the function must be greater than or equal to -1. Here is an example of such a function:
f(x) =
{ x² - 1, if -3 < x < 2 }{ x - 1, if x > 2 }
The choice of x² - 1 for the first interval is to ensure the output is greater than -1, and a simple linear function x - 1 for the second interval satisfies the condition that the output is greater than -1 as well. It's important to note that at x = 2, there's a break in the function, consistent with the domain specified.