Question

1. draw a function that has a domain of x ∈ ( - ∞,∞), but has a range equal to only one number.
2. explain why the single function on the left cannot be a valid function, while the two individual functions pictured in different colors on the right can each be a function.

reasoning:

Explanation:

Step1: Draw the function for question 2

A horizontal line \(y = c\) (where \(c\) is a constant) has a domain of \(x\in(-\infty,\infty)\) and a range of just the single - value \(c\). For example, \(y = 3\).

Step2: Recall the vertical - line test for question 3

A relation is a function if for every \(x\) - value in the domain, there is exactly one \(y\) - value. The left - hand circle fails the vertical - line test. A vertical line can intersect the circle at two points, meaning for some \(x\) values, there are two \(y\) values.

Step3: Analyze the right - hand functions

The two semi - circles on the right each pass the vertical - line test. For each \(x\) value in their respective domains, there is exactly one \(y\) value.

Answer:

1. Draw a horizontal line such as \(y = 3\).
2. The single function (circle) on the left is not a valid function because it fails the vertical - line test (a vertical line can intersect it at two points, so for some \(x\) values there are two \(y\) values). The two individual functions (semi - circles) on the right are functions because each passes the vertical - line test (for each \(x\) value in their domains, there is exactly one \(y\) value).