A function is a relation between two sets of data where each input is associated with one output. In other words, inputting a specific value into a function outputs one specific value.

Later, you will learn to use variables (like *x* and *y*) to write functions. For now, it’s important only to understand how a function table works with input/output. Let’s take a look at the following function table:

You can see that there is a consistent pattern for both the input data and the output data. The first input-output is 1-5. You can see that the output is 4 larger than the input. Then, as the input increases by 1, the output also increases by 1. Thus, the next rows are 2-6, 3-7, 4-8, and 5-9.

If you knew that the next input was 6, then could you guess what the output would be? Yes! The output must be 10.

Not only can you see the pattern and guess what future values will be, but you can also formally denote this in an equation. Since the output is always 4 more than the input, you can write the input/output equation as: 𝚫 + 4 = ⧠.

Let’s look at a few more examples. In the table below, see if you can find the missing values in the table.

For the input data column, the values increase by 1 in each row, so the missing value 𝚫 must be 7.

To find the output, you must understand the pattern. You can count the difference between each of the outputs, as the input increases by one step each time. The difference between 9 and 11 is 2, and the difference between 15 and 17 is 2 as well. Therefore, the input of 6 will have an output (⧠) of 13, which is 2 larger than 11 and 2 less than 15.

In this next example, let’s see if you can write an equation relating the input and output.

The input (2, 5, 6, 10, 13) seems quite random, so instead try to figure out how the output relates to the input. In the first row, the output (6) is 4 larger than the input (2). However, in the next row, the output (15) is 10 larger than the input (5). So addition doesn’t seem to explain the relationship.

Instead, let’s consider multiplication. In the first row, the output (6) is 3 times larger than the input (2). In the second row, the output (15) is once again 3 times larger than the input (5). In the third row, the output (18) is 3 times larger than the input (6). Now that you can see the pattern, write an equation showing how it works. The equation should show that the output (⧠) is equal to 3 times the input (𝚫). So, it should like: 𝚫 × 3 = ⧠.

**Answers to Practice Problems**

- Question 1
- 𝚫: 10 and ⧠: 28
- 𝚫 × 4 = ⧠
- Input: 16 | Output: 64

- Question 2
- 𝚫: 18 and ⧠: 22
- 𝚫 + 8 = ⧠

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