Unit 3: Functions & Linear Equations Homework 1: Relations & Functions Name: Bell:_ Date: “* This is a 2-page document! Find the domain and range, then represent as a table, mapping, and graph. 1/(-5,4), (4,-1), (-2, 1), (O, 4), (1, 3)} Domain Range = Domain s Range = Determine the domain and range of the following continuous graphs. 3. Domain Range 5. Domain- Range 6. Domain- Domain Range = Ranges ?
The correct answer and explanation is:
Let’s go step by step.
1. Finding the Domain and Range of the Given Set of Points
The given set of points is: (−5,4),(4,−1),(−2,1),(0,4),(1,3)(-5,4), (4,-1), (-2,1), (0,4), (1,3)
- Domain: The domain consists of all the x-values in the set. {−5,4,−2,0,1}\{-5, 4, -2, 0, 1\}
- Range: The range consists of all the y-values in the set. {4,−1,1,4,3}\{4, -1, 1, 4, 3\} Since 4 appears twice, we only list it once.
Thus:
- Domain = {−5,−2,0,1,4}\{-5, -2, 0, 1, 4\}
- Range = {−1,1,3,4}\{-1, 1, 3, 4\}
2. Representing the Data in Different Forms
Table Representation
| x | y |
|---|---|
| -5 | 4 |
| 4 | -1 |
| -2 | 1 |
| 0 | 4 |
| 1 | 3 |
Mapping Representation
We can represent the mapping as follows: Domain (x-values)Range (y-values)−544−1−210413\begin{array}{c|c} \textbf{Domain (x-values)} & \textbf{Range (y-values)} \\ -5 & 4 \\ 4 & -1 \\ -2 & 1 \\ 0 & 4 \\ 1 & 3 \\ \end{array}
In a visual mapping, each x-value is linked to its corresponding y-value.
Graph Representation
I’ll generate a scatter plot of these points.
Here is the graph representation of the given points. Each point is plotted in the coordinate system, and labeled accordingly.
3. Explanation (300 words)
A relation is a set of ordered pairs (x,y)(x, y), where each xx-value is associated with a corresponding yy-value. In the given problem, we have a set of five ordered pairs: (−5,4),(4,−1),(−2,1),(0,4),(1,3)(-5,4), (4,-1), (-2,1), (0,4), (1,3). From this, we can determine two key properties: the domain and range.
- The domain consists of all unique xx-values in the relation. Here, the domain is {−5,−2,0,1,4}\{-5, -2, 0, 1, 4\}.
- The range consists of all unique yy-values in the relation. Here, the range is {−1,1,3,4}\{-1, 1, 3, 4\}, noting that the value 4 appears twice but is listed only once.
To visually represent this relation, we use different formats:
- Table Representation organizes the values in a structured way, showing which xx-values correspond to which yy-values.
- Mapping Representation links each element in the domain to an element in the range using arrows.
- Graph Representation plots the points on a Cartesian plane, which helps in visualizing the relationship between the variables.
Understanding relations is fundamental in mathematics, as they form the basis for functions. A function is a special type of relation where each input (xx-value) has exactly one output (yy-value). In this case, since no xx-value is repeated with different yy-values, the given relation is also a function.
The ability to determine the domain and range, as well as represent a relation graphically, is crucial in analyzing mathematical models, data sets, and real-world problems. This knowledge is widely applied in disciplines such as physics, engineering, and economics.
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