Math Problem Statement
Identify the domain and range. Graph/plot the Points that represent the relation. Identify if the relation is function or not
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{(3, 0)(2,1) (1,2). (0.5), (1,4),(2,3), (3,2)}
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{(-2, 3),(-1,3), (-1,3), (0, 3), (1,3), (1.3)} 1
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y=2x+1 x={-1,-2, 0,1,2}
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Solve for y=x² {X = -3,-2,-1,0,1,2, 3}
Solution
Let's break down each part of the problem one by one:
1. Relation: {(3, 0), (2, 1), (1, 2), (0.5, 1), (1, 4), (2, 3), (3, 2)}
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Domain: The set of all x-values.
Domain = {3, 2, 1, 0.5}
Notice that some x-values repeat. -
Range: The set of all y-values.
Range = {0, 1, 2, 3, 4}
Since all the y-values are different or repeat multiple times. -
Graph: If we plot the points, we get the following points:
(3, 0), (2, 1), (1, 2), (0.5, 1), (1, 4), (2, 3), (3, 2) -
Is it a function?
A relation is a function if every x-value maps to exactly one y-value.
In this relation, some x-values (1, 2, 3) map to more than one y-value.
Therefore, this relation is not a function.
2. Relation: {(-2, 3), (-1, 3), (-1, 3), (0, 3), (1, 3), (1, 3)}
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Domain:
Domain = {-2, -1, 0, 1}
The set of distinct x-values. -
Range:
Range = {3}
All y-values are 3. -
Graph: Points are:
(-2, 3), (-1, 3), (0, 3), (1, 3) -
Is it a function?
A relation is a function if each x-value has exactly one y-value.
Here, every distinct x-value has the same y-value (3).
Therefore, this relation is a function.
3. Relation: ,
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Domain:
The given domain is . -
Range:
We calculate the y-values for each x:- When ,
- When ,
- When ,
- When ,
- When ,
Range = {-3, -1, 1, 3, 5}
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Graph: Points are: (-1, -1), (-2, -3), (0, 1), (1, 3), (2, 5)
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Is it a function?
Yes, because for every x-value there is exactly one y-value.
Therefore, this relation is a function.
4. Relation: ,
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Domain:
The given domain is . -
Range:
We calculate the y-values for each x:- When ,
- When ,
- When ,
- When ,
- When ,
- When ,
- When ,
Range = {0, 1, 4, 9}
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Graph: Points are: (-3, 9), (-2, 4), (-1, 1), (0, 0), (1, 1), (2, 4), (3, 9)
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Is it a function?
Yes, because for every x-value there is exactly one y-value.
Therefore, this relation is a function.
Summary:
- Not a function: Repeated x-values with different y-values.
- Function: Each x-value has one y-value.
- Function: Linear equation with distinct x-values.
- Function: Quadratic equation with distinct x-values.
Would you like a more detailed explanation or graph for any of these parts?
Follow-up Questions:
- How do we determine the domain and range of more complex functions?
- What is the graphical representation of a function?
- What makes a relation not a function?
- How can we use the vertical line test to identify functions?
- What are some examples of quadratic functions?
Tip:
Always remember that a relation is a function if and only if each x-value corresponds to exactly one y-value. This can be visually checked using the vertical line test on a graph.
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Math Problem Analysis
Mathematical Concepts
Functions
Relations
Domain and Range
Linear Equations
Quadratic Equations
Formulas
y = 2x + 1
y = x^2
Theorems
Function: Each input (x) must map to one and only one output (y).
Vertical Line Test: A graph represents a function if no vertical line intersects it at more than one point.
Suitable Grade Level
Grades 8-10