Math Problem Statement

Identify the domain and range. Graph/plot the Points that represent the relation. Identify if the relation is function or not

  1. {(3, 0)(2,1) (1,2). (0.5), (1,4),(2,3), (3,2)}

  2. {(-2, 3),(-1,3), (-1,3), (0, 3), (1,3), (1.3)} 1

  3. y=2x+1 x={-1,-2, 0,1,2}

  4. 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)}

  • 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)}

  • 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: y=2x+1y = 2x + 1, x={1,2,0,1,2}x = \{-1, -2, 0, 1, 2\}

  • Domain:
    The given domain is x={1,2,0,1,2}x = \{-1, -2, 0, 1, 2\}.

  • Range:
    We calculate the y-values for each x:

    • When x=1x = -1, y=2(1)+1=2+1=1y = 2(-1) + 1 = -2 + 1 = -1
    • When x=2x = -2, y=2(2)+1=4+1=3y = 2(-2) + 1 = -4 + 1 = -3
    • When x=0x = 0, y=2(0)+1=1y = 2(0) + 1 = 1
    • When x=1x = 1, y=2(1)+1=3y = 2(1) + 1 = 3
    • When x=2x = 2, y=2(2)+1=5y = 2(2) + 1 = 5

    Range = {-3, -1, 1, 3, 5}

  • Graph: Points are: (-1, -1), (-2, -3), (0, 1), (1, 3), (2, 5)

  • Is it a function?
    Yes, because for every x-value there is exactly one y-value.
    Therefore, this relation is a function.

4. Relation: y=x2y = x^2, x={3,2,1,0,1,2,3}x = \{-3, -2, -1, 0, 1, 2, 3\}

  • Domain:
    The given domain is x={3,2,1,0,1,2,3}x = \{-3, -2, -1, 0, 1, 2, 3\}.

  • Range:
    We calculate the y-values for each x:

    • When x=3x = -3, y=(3)2=9y = (-3)^2 = 9
    • When x=2x = -2, y=(2)2=4y = (-2)^2 = 4
    • When x=1x = -1, y=(1)2=1y = (-1)^2 = 1
    • When x=0x = 0, y=02=0y = 0^2 = 0
    • When x=1x = 1, y=12=1y = 1^2 = 1
    • When x=2x = 2, y=22=4y = 2^2 = 4
    • When x=3x = 3, y=32=9y = 3^2 = 9

    Range = {0, 1, 4, 9}

  • Graph: Points are: (-3, 9), (-2, 4), (-1, 1), (0, 0), (1, 1), (2, 4), (3, 9)

  • Is it a function?
    Yes, because for every x-value there is exactly one y-value.
    Therefore, this relation is a function.

Summary:

  1. Not a function: Repeated x-values with different y-values.
  2. Function: Each x-value has one y-value.
  3. Function: Linear equation with distinct x-values.
  4. Function: Quadratic equation with distinct x-values.

Would you like a more detailed explanation or graph for any of these parts?

Follow-up Questions:

  1. How do we determine the domain and range of more complex functions?
  2. What is the graphical representation of a function?
  3. What makes a relation not a function?
  4. How can we use the vertical line test to identify functions?
  5. 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