write a piece-wise function over 3 intervals whose domain is (-infinity, infinity) and whose range is (-infinity,-3) united (1,10). include an increasig, decreasing, and a constant function.
Coming up with these is difficult, but working with these is not. Do you know what the domain and the range are, for starters? Have you ever seen this notation before? I have never seen the word united in this sense before. Do you know how to determine when a function is increasing, decreasing, or remaining the same?
To create a piece-wise function over three intervals with the given domain and range, we can divide the domain into three parts: (-∞, a), (a, b), and (b, ∞). Where 'a' and 'b' are the values that will separate the intervals.
Let's start by creating an increasing function in the first interval (-∞, a):
For the interval (-∞, a), we want the function to increase from -∞ to -3. A linear function can achieve this. Let's define the first interval function as g(x) = -3 - x.
Now, let's create a decreasing function in the second interval (a, b):
For the interval (a, b), we want the function to decrease from -3 to 1. A different linear function can accomplish this. Let's define the second interval function as h(x) = 4 - x.
Lastly, we need a constant function in the third interval (b, ∞):
For the interval (b, ∞), we want the function to have a constant value of 10. Therefore, we define the third interval function as f(x) = 10.
Combining all the three interval functions, we get the piece-wise function:
f(x) = { -3 - x, for x < a
4 - x, for a < x < b
10, for x > b }
The domain of this function is (-∞, ∞), and the range is (-∞, -3) U (1, 10).