find the domain of the following piecewise function. f(x) 3x-2 if -1 2x+3 if 0
The Correct Answer and Explanation is:
Let’s first define the piecewise function and understand how to find its domain.
The function is given as: f(x)={3x−2if x≤−12x+3if x>0f(x) = \begin{cases} 3x – 2 & \text{if } x \leq -1 \\ 2x + 3 & \text{if } x > 0 \end{cases}f(x)={3x−22x+3if x≤−1if x>0
Finding the Domain:
The domain of a function is the set of all possible values of xxx for which the function is defined.
In this case, we have two pieces to consider:
- The first piece, 3x−23x – 23x−2, is defined for x≤−1x \leq -1x≤−1. This means that this part of the function is valid for values of xxx less than or equal to −1-1−1.
- The second piece, 2x+32x + 32x+3, is defined for x>0x > 0x>0. This means that this part of the function is valid for values of xxx strictly greater than 0.
Domain of the Piecewise Function:
- The first part is valid for x≤−1x \leq -1x≤−1, so we include all values less than or equal to −1-1−1.
- The second part is valid for x>0x > 0x>0, so we include all values greater than 0.
The important thing to notice here is that there is no piece of the function defined for −1<x≤0-1 < x \leq 0−1<x≤0. Therefore, there is a gap between the two pieces of the function, meaning no value of xxx exists between −1-1−1 and 0 in this case.
Conclusion:
The domain of the piecewise function is: (−∞,−1]∪(0,∞)(-\infty, -1] \cup (0, \infty)(−∞,−1]∪(0,∞)
This means the function is defined for all xxx values that are either less than or equal to −1-1−1, or greater than 0. However, it is not defined for values in the interval (−1,0](-1, 0](−1,0].
