Question

What is the range of this piecewise function

Answer 1

The range of the piecewise function described, which is a horizontal line with a constant value of 20 for 0 ≤ x ≤ 20, is simply the single value {20}.

Step-by-step explanation:

To determine the range of the piecewise function described, we need to look at the output values that f(x) can take. Given the representation of f(x) as a horizontal line in the context of the function definition for f(x) being a constant value (in this case, 20), and considering the domain of 0 ≤ x ≤ 20, the range is simply the set of y-values that the function outputs within the specified x interval.

Since f(x) is a horizontal line and it is defined as a constant value of 20 over the domain 0 ≤ x ≤ 20, the function outputs the value 20 for any x in that interval. Therefore, the range of f(x) is a single value, which is {20}. There are no other output values for f(x), as x varies between 0 and 20, so the entire range of the piecewise function is {20}.

Answer 2

Answer as a compound inequality:
`-4 \le y < 2`

Answer in interval notation: [-4, 2)

=============================================

Step-by-step explanation:

The range is the set of all possible y outputs of a function. When dealing with a graph like this, we just look at the highest and lowest points to determine which y values are possible.

The lowest point occurs when y = -4. We include this value. So far we have
`y \ge -4` which is the same as
`-4 \le y`

The upper ceiling for the y value is y = 2. We can't actually reach this value because of the open hole at (-3,2). So we say that
`y < 2`

Combine
`-4 \le y` and
`y < 2` to get the compound inequality
`-4 \le y < 2`

This says y is between -4 and 2, including -4 but excluding 2.

To convert this to interval notation, we write [-4, 2) where the square bracket says to include the endpoint and the curved parenthesis says to exclude the endpoint.