# Explainer: Linear Functions

In this explainer, we will learn how to identify, write, and evaluate a linear function and complete its function table.

Let’s recall what a function is: it is a type of relation between two sets of objects, called the input and the output, where each input value is associated with exactly one output value.

The input and output can be sets of numbers, and the relation between the input and the output is then often described with an equation, called the equation of a function.

### Equation of a Function

The equation of a function describes the mathematical relationship between the output value, called (read “ of ”) or , and the input value (often called ).

Let’s take as an example the relation that assigns to any number its half. So, for any value of , the associated value is or . The function can be seen as a machine that takes as input a value and gives as output .

### Function Value for a given Input

The output of a function for a given input is called the function value for this input.

When the equation of a function is given, it is easy to find the value of the function, that is, the output of the function, for any given input, since the equation “tells” us how to calculate it.

For instance, in the previous example of , if the value of is 6, then the function value is , that is, 3. If the input is , then the output is given by , that is, . We write and .

We have found here two ordered pairs that correspond to the function : and .

### How to Find a Function Value for a given Input

Write the equation and substitute the given input for in it. Then, carry out the whole calculation. The function value is the final result of this calculation.

When one has to find the function value for several inputs, it is very helpful to write the obtained ordered pairs in a table, called a function table, where the first row is for the input values, and the second row shows the output value found for each of them using the equation.

Let’s go through some examples.

### Example 1: Finding a Function Value for a given Input

Complete the given function table.

Input Output
9 ?
5 ?
16 ?

The table gives three different input values: 9, 5, and 16, and we are asked to find the output values given by , where is the input value.

Let’s start with 9: we substitute 9 for in , which gives . Hence, when the input is 9, the output is 11.

With an input of 5, the output is .

For an input of 16, the output is .

Input Output
9 11
5 7
16 18

### Example 2: Finding a Function Value for a given Input

Complete the given function table.

Input Output
2 ?
7 ?
4 ?

The table gives three different input values: 2, 7, and 4, and we are asked to find the corresponding output values given by , where is the input value.

For an input of 2: we substitute 2 for in , which gives . Hence, the output is 6.

We do the same for 7: . An input of 7 gives an output of 21.

And for 4, . An input of 4 gives an output of 12.

Input Output
2 6
7 21
4 12

### Example 3: Finding a Function Value for a given Input

Find the value of given the function .

The equation of the function , , tells us that every output is found by multiplying the input by 7 and then subtracting this value from 3. We want to find the value of , that is, when . For this, we simply substitute 8 for in the equation in order to perform the calculation with 8 as input. It gives

### Example 4: Finding a Function Value for a given Input

Fill in the input-output table for the function .

InputOutput
0
2
4
5

The equation of the function tells us that every output is found by multiplying the input by 5 and then adding 3. For any given value of , we simply need to substitute the given value for in the equation to perform this calculation to find the value of the output .

For 0, we find . For an input 0, the output is 3.

For 2, we find . When the input is 2, the output is 13.

For 4, we find . When the input is 4, the output is 23.

And, finally, for 5, we find . The missing value under 5 in the table is 28.

Input Output
0 3
2 13
4 23
5 28

We have learned how to find a function value for a given input when we have the equation of the function. Now, for a simple function, it is possible to find its rule, that is, its equation, when we have its function table. Let’s see with one example how it goes.

### Example 5: Finding the Function Rule from a Function Table

Find the rule for the given function table.

Input Output
1 9
4 12
10 18