What is the function rule for input3, output3, input4, output5, input6, output9

To determine the function rule for the given input and output values, we need to identify a pattern or relationship between the inputs (input3, input4, input6) and their corresponding outputs (output3, output5, output9).

Let's examine the input and output values:

Input Output
3 Output3
4 Output5
6 Output9

From the given information, we can observe that when the input value is increased by 1, the output value increases by an odd number. Specifically,

- input 4 (3+1) corresponds to output 5 (Output3+2 = Output5),
- input 6 (4+2) corresponds to output 9 (Output5+4 = Output9).

Based on this pattern, we can infer that the function rule involves adding an increasing odd number to each input value to produce the corresponding output value.

To find the specific pattern, we need to calculate the difference between each output value and its respective input value:

Output3 - Input3 = 0
Output5 - Input4 = 1
Output9 - Input6 = 3

By examining the differences, we can see that they form a pattern: 0, 1, 3.

This suggests that the function rule involves adding increasing odd numbers starting from 0.

Therefore, the function rule can be expressed as:

Output = Input + (2n + 1)

Substituting values from our examples:

Output3 = Input3 + (2n + 1)
Output3 = 3 + (2(0) + 1)
Output3 = 3 + 1
Output3 = 4

Output5 = Input4 + (2n + 1)
Output5 = 4 + (2(1) + 1)
Output5 = 4 + 3
Output5 = 7

Output9 = Input6 + (2n + 1)
Output9 = 6 + (2(3) + 1)
Output9 = 6 + 7
Output9 = 13

Therefore, the function rule for the given input and output values is Output = Input + (2n + 1), where n represents the difference in position between each input and output pair.