the graph of the relation s is shown below
Relation \(S\) is represented in the coordinate plane by a curved piece, a vertical segment, and one isolated point. The domain consists of all \(x\)-values that appear on the graph, and the range consists of all \(y\)-values that appear on the graph.
Assumed mathematical description of S
A concrete interpretation consistent with the displayed graph is the union of three pieces:
\(S = \{(x,(x+2)^2-1)\,:\,-4 \le x \le 0\}\ \cup\ \{(2,y)\,:\,-2 \le y \le 2\}\ \cup\ \{(4,3)\}.\)
Domain of S
The domain is the set of all \(x\) for which at least one point \((x,y)\) lies in \(S\). The curved piece contributes all \(x\) from \(-4\) through \(0\), the vertical segment contributes \(x=2\), and the isolated point contributes \(x=4\). Therefore, \[ \mathrm{Dom}(S) = [-4,0]\ \cup\ \{2,4\}. \]
Range of S
The range is the set of all \(y\) for which at least one point \((x,y)\) lies in \(S\). On the curved piece \(y=(x+2)^2-1\) for \(-4 \le x \le 0\), the minimum occurs at the vertex \(x=-2\), giving \(y=-1\), and the maximum on that interval is \(y=3\) (at \(x=-4\) and \(x=0\)). The vertical segment at \(x=2\) covers all \(y\) from \(-2\) to \(2\), and the isolated point adds \(y=3\) (already included). Therefore, \[ \mathrm{Ran}(S) = [-2,3]. \]
Function status
A function assigns at most one \(y\)-value to each \(x\)-value. The vertical segment at \(x=2\) contains multiple points \((2,y)\) with different \(y\)-values (for example \((2,-1)\) and \((2,1)\)), so the vertical line \(x=2\) intersects the graph more than once. The relation does not represent a function.
| Property | Result | Graph-based justification |
|---|---|---|
| Domain | \(\,[-4,0]\cup\{2,4\}\,\) | All \(x\)-values used by the curved piece, the vertical segment, and the isolated point |
| Range | \(\,[-2,3]\,\) | Vertical segment reaches down to \(-2\); curved piece and isolated point reach up to \(3\) |
| Function | No | At \(x=2\), multiple \(y\)-values occur (vertical line test fails) |
Common pitfalls
- Endpoint inclusion: filled endpoints indicate inclusion of boundary values in intervals such as \([-4,0]\) and \([-2,2]\).
- Domain vs. range: horizontal coverage corresponds to the domain, vertical coverage corresponds to the range.
- Function test confusion: a graph can contain curved pieces and still fail to be a function if any vertical line intersects it more than once.