Strategy for solving functions
1. Make sure you have consecutive inputs.
Example:
Input
3
4 output
12
15
6
7
17
18
In this example you do not have consecutive inputs.
2. If the outputs go up as the input goes up, then you have a direct proportion or direct variation. Look out the difference in the outputs.
A. If the difference in the outputs is the same as the difference in the inputs, then the operation is addition or subtraction.
Input
5 output
17
18 6
7
8
19
20
In this example, the difference in the outputs is the same as the inputs, so the operation is addition or subtraction.
If the input was N, the output would be N + 12
B. If the difference in the outputs is constant, then the operation is multiplication or division.
Input output
9
10
11
17
21
25
12 29
In this example, the difference in the outputs 4, so the operation is multiplication, and 4N is part of the solution.
Make a 4N column and continue looking for a pattern.
Input 4 x input output
9 36 17
10
11
12
N
40
44
48
4N
21
25
29
4N-19 is the solution
C. If the difference in the outputs is consecutive even or odd numbers, then you must factor the outputs and look for a pattern, relating each factor to the inputs.
Input output
3
4
5
6
The difference in the outputs is 12, then 14, then 16, so you must factor input output factors
3 30 30x1, 15x2, 10x3, 6x5 (if you start with the larger factor, then continue using the larger factor
4
30
42
56
72
42 first in order to identify a pattern)
21x2, 14x3, 13x4, 7x6 (Since I used 1 with the first output,
I don’t need to use it with the second output.)
5
6
56
72
14x4, 8x7
12x6, 9x8
Now look for a pattern: 6x5, 7x6, 8x7, 9x8
N (N+3)(N+2)

D. If the difference in the outputs is consecutive numbers, then you must double the output, then factor and look for a pattern, relating each factor to the inputs. Don’t forget to add divide by 2 in you answer. input
3
4 output
5
9
5
6
14
20
Difference in the outputs is 4, then 5, then 6
Since your output differences are consecutive numbers, you must double our output then factor. input output 2 x output factors
3
4
5
5
9
14
10
18
28
10x1, 5x2
9x2, 6x3
7x4
6
N
20 40
Therefore the solution to this problem is:
N (N+2)(N-1)
8x5
(N+2)(N-1)
2
3. If the outputs go down as the input goes up, then you have an indirect proportion or inverse variation. You must look for a pattern relating the input to the output.
Input
1
2
3 output
12
11
10
4 9
This is an inverse proportion because the outputs go down as the inputs go up, so you look at doing something to each input and its output together. If you add the input to the output, then you get:
Input output sum
1
2
3
4
12
11
10
9
13
13
13
13
Each time the sum is 13, thus output = 13 – input, so the formula is:
N 13-N
Another example:
Input
1
2
3 output
36
18
12
4 9
In this situation, you must multiple input by output, and you get 26 each time, thus:
N 36/N