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The domain are a set of input values and the range are a set of output values.

The domain is all values that are allowed for the independent variable (x-value).
The range is all the output variable (f(x)) when the value of x is evaluated.
Both describe the x & y values. 
An equation can be linear, absolute, exponential, or square-root . 
There are different ways of obtaining the domain & range.
This includes discovering by points, creating tables and visualizing a graph.
It can be discovered  by changing the function into its' inverse. (arriving soon)
DOMAIN & RANGE
Examples and differennt ways of discovering the domain and range
Coordinate pairs (x, y): 
(0,-2), (2, 3), (3, 5), (-2, 0), (-3, -2), (-5, -3)
Domain:  {0, 2, 3, -2, -3, -5}
Range: {-2 ,3, 5, 0,-2, -3}
Steps and ways of discovering the domain & range
If it is linear, then:
  • Create a table
  • Discover initial points (-x or y-intercept, factoring or quadratic equation formula)
  • Discover additional points
  • Create a graph 
  • Define the boundary of the x or y-values.
  • Discover the inverse function. (arriving soon)
  • Write the domain & range notation form.
If it is NOT linear, then:
  • Find the vertex 
  • Create a table
  • Discover initial points (roots, y-intercept, factoring or quadratic equation formula)
  • Discover additonal points
  • Create a graph 
  • Define the boundary of the x or y-values.
  • Discover the inverse function. (arriving soon)
  • Write the domain & range notation form.
Linear Function: Domain & Range
linear function: f(x) = x - 3

  • Is it linear? y = x - 3  Yes. We know that it is linear because it doesn't have absolute values, even exponent values, or square root values.

  • Create a table
  • Discover initial points (-x or y-intercept, factoring or quadratic equation formula)
  • Discover additional points​

0

-4

(-1, -4)

1

(4 , 1)

f(x)

-3

0

(0, -3) -- y-intercept

(3, 0) -- x-ntercept

  x, y

(1, -2)

x

x - 3 = y

0

3

0 - 3  = -3

3 - 3  = 0

1

-1

-1 - 3  = -4

4

4 - 3  =  1

 1 - 3  =  -2

  • Create a graph from the points.
         -- place the x-y intercept
         -- place the other points

10

-2

6

(-1, -4)

-10

-8

-6

-4

-2

(0, -3)

10

(4, 1)

6

8

4

(1, -2)

4

2

(3. 0)

  • Determine the boundary of the x or y-values.

​Any number can be substituted for x, in the table.

Therefore, the boundary limits are infinite.
For every infinite x-values (domain) there is infinite y-values (range).

  • Discover the inverse function. (arriving soon)

  • Write the domain & range notation form.

||R = all real numbers = 
Domain: x | x      ||R
C
(     , -     )
 O
O
 O
O
Range: y | y      ||R
C
Absolute Value: Doman & Range
linear function:  f(x) = | x - 3 |

  • Is it linear?  y = | x - 3 |; No. This function has a absolute value. Therefore, we must discover the vertex, first.

  • Find the vertex 

y = a | x + h | + k
y = (x - 3) + 0

h = 3
k = 0

Vertex Coordinate: (h, k) = (x, y) = (3 , 0)
  • Discover initial points (roots, y-intercept, factoring or quadratic equation formula)
Substitute, various values for x

x

| x - 3 |

f(x)

  x, y

  (0, 3) 

  (3, 0)  --vertex

(6, 3)

3

0

 0

3

  | 0 - 3 |  

   | 3 - 3 |  

 7

3

6

| 6 - 3 |

-4

| -4 - 3 |

  (-4, 7)

  • Discover the direction of the absolute value

Although, we have the points and can graph the equation, we need to discover the direction.

We can utilize the vertex form:
Substitute vertex (3, 0) for (h, k) and any known point on the graph (-4,  7) for (x, y)
   
y = a | x + h | + k
7 = a | -4 - 3 | + 0
7 = a | -7 | + 0 --absolute vale | -7 | = 7
7  =    7a

     1 = a

a = +1; a > 0;  direction-- UP

7           7  

  • Create a graph from the points.

-4

10

8

(3, 0)

6

4

2

-2

-10

-8

-6

-4

-2

8

(-4,  7)

6

2

(0, 3)

4

(6, 3)

  • Determine the boundary of the x or y-values.


VERTEX: The h = x = 3; k = y = 0 is the minimum value --> (vertex (3, 0))  
TABLE: The boundary limits are @ x = 3; y = 0 
GRAPH: The graph displays the linear lines pointing upward from the vertex point (3, 0) where, x = 3 and y = 0 (3, 0). Therefore, any value is >  3 and < 3 on the x-axis for all real numbers, hence (-     ,         ).
All values of y must start at zero, and continue upward on te y-axis for all x-value until infinity, hence [ 0,        ]. 
 O
O
 O
O
 O
O

  • Discover the inverse function. (arriving soon)

  • Write the domain & range notation form.

Domain: x | x       (-     ,    ) 
C
 O
O
 O
O
Range: y | y        y > 0  [0,     )
 O
O
C
Parabola:  Domain & Range
linear function: f(x) = (x  - 4)

2

  • Is it linear?                                ;  No.  Parabola do not have straight lines but curves. This function has an exponent value raised to the 2nd power. Therefore, we must discover the vertex.

y =    x   - 4

2

  • Find the vertex
Discover the vertex from standard form

2

2

standard form: y =  Ax    - Bx   +  C   -->    x   - 0x  - 4

A =  1
B =  0
C =  -4

2

x
=
=
(    )
 b 
 2a
=
(     )
   0   
 2(1)
=
(    )
  0  
  2
0

2

Substitute 0 for x:
y  =  x   +  0x  - 4
y  =  (0)   +  0(0)  -  4
y  =  0  +  0  - 4
y  =  -4

2

Vertex Coordinate: (h, k), (x, y) = (0 , -4)
  • Discover initial points (roots, y-intercept, factoring or quadratic equation formula)

2

Discover the root by factoring
x   +  0x  - 4
(x - 2)(x + 2)
x + 2

​       x = 2  &  x = -2
Root:  (2, 0) and (-2, 0)

  • Create a table of points
Substitute, various values for x

  (x, y)

(0, -4)

(2, 0)

(2, 0)

x

  x    - 4 

2

f(x)

-4

0

 0

2

-2

0

         0     -  4  = 0 - 4 = -4

2

12

-4

(0, 12)

         2     -  4  = 4 - 4 = 0

2

      -2     - 4  = 4 - 4 = 0

2

      -4     - 4  = 14 - 4 = 12

2

  • Discover the direction of the equation

Although, we have the points and can graph the equation, we need to discover the direction.

We can utilize the vertex form:
Substitute vertex (0, -4) and any known point on the graph (ex. 4, 12)

         y = a | x + h | + k
        12 = a(4 - 0)  - 4
        12 = a(4) - 4
12 + 4 = 4a -4 + 4
        16 = 4a
       16  = 4a

          4 = a

a = +4; a > 0;  direction-- UP

4      4  

  • Create a graph with points

6

8

-4

(0, -4)

6

2

(-2, 0)

(2, 0)

4

-8

-6

-4

4

-2

6

6

  • Determine the boundary of the x or y-values.


VERTEX: The h = x = 0; k = y = -4 is the minimum value --> (vertex (0, -4))  
TABLE: The boundary limits:  x = -2; x = -2; y = -4
GRAPH: The graph displays the linear lines pointing upward from the vertex point (0, -4) where, x = 0 and y = -4 (0, -4). Therefore, any value is >  0 on the x-axis for all real numbers, hence (-     ,         ).
All values of y must start at -4, and continue upward on the y-axis for all x-valuesa until infinity, hence
[ -4,        ]. 
 O
O
 O
O
 O
O

  • Discover the inverse function. (arriving soon)

  • Write the domain & range notation form.

Domain:  x | x  (-     ,       )
 O
O
 O
O
Range:   y | y  [ -4,     ) 
 O
O
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