Question
Not a function Function Not a function Relation 3 \[ \{(2,0),(2,-1),(2,-3),(2,2)\} \] Function Not a function Relation 4 \[ \{(z,-9),(m, 0),(u, 1),(j,-2)\} \] Function Not a function
Ask by Lang Hall. in the United States
Mar 09,2025
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Tutor-Verified Answer
Answer
Relation 3 is **not a function** because the input \(2\) is paired with multiple outputs.
Relation 4 is a **function** since each input is paired with exactly one output.
Solution
1. **Definition of a Function**
A relation is a function if and only if every input is associated with exactly one output.
2. **Analyzing Relation 3**
The relation is
\[
\{(2,0),\,(2,-1),\,(2,-3),\,(2,2)\}.
\]
- Here, the input \(2\) appears in every ordered pair, but it is associated with multiple different outputs: \(0\), \(-1\), \(-3\), and \(2\).
- Since the same input \(2\) has more than one output, the relation does not satisfy the definition of a function.
**Conclusion:** Relation 3 is **not a function**.
3. **Analyzing Relation 4**
The relation is
\[
\{(z,-9),\,(m,0),\,(u,1),\,(j,-2)\}.
\]
- Here, the first elements \(z\), \(m\), \(u\), and \(j\) serve as the inputs.
- Provided that \(z\), \(m\), \(u\), and \(j\) are distinct (which is the normal assumption), each input is paired with exactly one output.
**Conclusion:** Relation 4 is a **function**.
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Simplify this solution Beyond the Answer
In mathematics, a function is defined as a relation in which every input (or 'x' value) is associated with exactly one output (or 'y' value). In the first relation, all pairs share the same 'x' value of 2 but have multiple 'y' values, thus it can't be a function. In contrast, a relation where each input has a unique output can be classified as a function, as seen in the second relation. When grappling with functions, a common mistake is confusing pairs with unique outputs as a function despite repeated inputs. Always remember: for a set to be a function, no 'x' value should appear more than once! Double-checking your pairs can save a lot of confusion and frustration down the road.
