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EXAMPLES & SOLUTIONS
Anchor 1
It requires that you know how many input values are associated with each output value.
Functions are always linear.
​Not every equation is a function.
Creating a graph can assist in determining if a curve is a function or not.
Another technique is to simply observe a graph to determine if it is a function.
This method is called the vertical line test.
IDENTIFY FUNCTIONS
Explanation of how to identify one-to-one functions
EXAMPLE: LINEAR EQUATION
f(x) = 1x + 3
4
8
6
7
5
3
1
2
-1
-3
-2
-2
-4
-6
(-3,0)
(-5, -2)
-6
-8
-7
-9
-5
-4
(3, 6)
(5, 8)
(0, 3)
Equation
y = x + 3
8
6
4
Domain:
Range:
x-intercept: (-3, 0)
y-intercept: (0, 3)
(- , )
O
O
O
O
(- , )
O
O
O
O
Equation Table
y
3
0
-2
x
input association output
f(x) = 1x + 3
1(0) + 3
1(-3) + 3
1(3) + 3
1(-5) + 3
0
-3
1(5) + 3
8
3
6
-5
5
Functions can have one x-input for every y-output
ONE - to - ONE
0
-3
3
5
-5
3
0
6
8
-2
IDENTIFY FUNCTION
Explanation of how to identify many-to-one functions
EXAMPLE: ABSOLUTE EQUATION
f(x) = | x | + 1
Equation
(2, 3)
(4, 5)
y = | x | + 1
-3
-6
-8
-7
-10
-9
-5
-4
8
6
4
4
(0, 1 )
8
6
7
5
1
2
3
-1
-2
-2
-4
-6
(-4, 5)
(-2, 3)
Domain:
Range:
vertex: (0, 1)
( - , )
O
O
O
O
[ 1, )
O
O
Equation Table
y
1
3
3
5
-4
| -4 | + 1
5
| 4 | + 1
4
x
input association output
f(x) = | x | + 1
| 0 | + 1
| 2 | + 1
| -2 | + 1
0
2
-2
Functions can have many inputs to one output
MANY - to - ONE
0
2
-2
4
-4
1
3
5
Fact: To consider a function a function, each x-value can have only one y-value
IDENTIFY NON-FUNCTIONS
Explanation of how to identify one-to-many functions
EXAMPLE: PARABOLA
x = y
2
8
6
7
8
5
(4, -2)
3
4
-6
-7
-4
6
(4, 2)
(1, 1)
1
(1, -1)
-1
-2
-2
-3
-5
-4
2
(0, 0 )
2
y = x
y = - x
Equation
x = y
2
Domain:
Range:
x-intercept: (0, 0)
y-intercept: (0,0)
(- , )
O
O
O
O
[0, )
O
O
Equation Table
y
0
1
x
input association output
0
x = y = y
2
0
+ 1
4
+ 2
+ 1
+ 4
Functions CANNOT have one input to many outputs
ONE- to - MANY
0
2
4
0
1
-1
-2
2
IDENTIFY NON-FUNCTIONS
Explanation of how to identify many-to-many
EXAMPLE: EXPONENT (SQUARE ROOT)
x + y = 25
2
2
8
7
6
5
4
1
2
3
(3, -4)
(0, -5)
-6
(-3, -4 )
-4
-3
-4
-1
-2
-2
-6
-8
-7
-10
-9
-5
(5, 0)
Equation
x + y = 25
2
2
(-5, 0)
(-3, 4)
6
(3, 4)
(0, 5)
2
4
Domain:
Range:
x-intercept:
y-intercept:
( -5, 0), (5, 0)
(0, -5), (0, 5)
| -5, 5 | , -5 < x < 5
| -5, 5 | , -5 < y < 5
Equation Table
-3
-3 + y = 25
2
2
(-3)(-3) + y = 25 - 9
2
9 - 9 + y =
2
16
+ 4
=
y
2
16
-5
(-5)(-5) + y = 25 - 5
2
-5 + y = 25
2
2
25 - 25 + y =
2
25-25
=
y
2
0
0
5
-5 + y = 25
2
2
0
0
5 - 5 + y =
2
25
5 - 5 + y =
2
25
=
y
2
25
(5)(5) + y = 25 - 5
2
25 - 25 + y =
2
25-25
=
y
2
0
0 + y = 25
0 + y = 25 - 9
2
+ 5
3 + y = 25
2
2
( 3)( 3) + y = 25 - 9
2
+ 4
y
9 - 9 + y =
2
16
=
y
2
16
3
x + y = 25
2
2
input association output
x
Functions CANNOT have many inputs to the many outputs
MANY to MANY
0
-3
3
5
-5
-5
5
4
-4
0
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