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 𝑓(π‘₯) (read β€œπ‘“ of π‘₯”) or 𝑦, and the input value (often called π‘₯).

Let’s take as an example the relation that assigns to any number its half. So, for any value of π‘₯, the associated value is 𝑓(π‘₯)=π‘₯2 or 𝑦=π‘₯2. The function can be seen as a machine that takes as input a value π‘₯ and gives as output π‘₯2.

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 β€œtells” us how to calculate it.

For instance, in the previous example of 𝑓(π‘₯)=π‘₯2, if the value of π‘₯ is 6, then the function value is 62, that is, 3. If the input is βˆ’19, then the output is given by βˆ’192, that is, βˆ’9.5. We write 𝑓(6)=3 and 𝑓(βˆ’19)=βˆ’9.5.

We have found here two ordered pairs that correspond to the function 𝑓: (6,3) and (βˆ’19,βˆ’9.5).

How to Find a Function Value for a given Input

Write the equation and substitute the given input for π‘₯ 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 (π‘₯)Output (2+π‘₯)
9 ?
5 ?
16 ?

Answer

The table gives three different input values: 9, 5, and 16, and we are asked to find the output values given by 2+π‘₯, where π‘₯ is the input value.

Let’s start with 9: we substitute 9 for π‘₯ in 2+π‘₯, which gives 2+9=11. Hence, when the input is 9, the output is 11.

With an input of 5, the output is 2+5=7.

For an input of 16, the output is 2+16=18.

Input (π‘₯)Output (π‘₯+2)
9 11
5 7
16 18

Example 2: Finding a Function Value for a given Input

Complete the given function table.

Input (π‘₯)Output (3π‘₯)
2 ?
7 ?
4 ?

Answer

The table gives three different input values: 2, 7, and 4, and we are asked to find the corresponding output values given by 3π‘₯, where π‘₯ is the input value.

For an input of 2: we substitute 2 for π‘₯ in 3π‘₯, which gives 3Γ—2=6. Hence, the output is 6.

We do the same for 7: 3Γ—7=21. An input of 7 gives an output of 21.

And for 4, 3Γ—4=12. An input of 4 gives an output of 12.

Input (π‘₯)Output (3π‘₯)
2 6
7 21
4 12

Example 3: Finding a Function Value for a given Input

Find the value of 𝑓(8) given the function 𝑓(π‘₯)=3βˆ’7π‘₯.

Answer

The equation of the function 𝑓, 𝑓(π‘₯)=3βˆ’7π‘₯, tells us that every output is found by multiplying the input π‘₯ by 7 and then subtracting this value from 3. We want to find the value of 𝑓(8), that is, when π‘₯=8. For this, we simply substitute 8 for π‘₯ in the equation in order to perform the calculation with 8 as input. It gives 𝑓(8)=3βˆ’7Γ—8𝑓(8)=3βˆ’56𝑓(8)=βˆ’53.

Example 4: Finding a Function Value for a given Input

Fill in the input-output table for the function 𝑦=5π‘₯+3.

InputOutput
0…
2…
4…
5…

Answer

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

For 0, we find 5Γ—0+3=0+3=3. For an input 0, the output is 3.

For 2, we find 5Γ—2+3=10+3=13. When the input is 2, the output is 13.

For 4, we find 5Γ—4+3=20+3=23. When the input is 4, the output is 23.

And, finally, for 5, we find 5Γ—5+3=25+3=28. The missing value under 5 in the table is 28.

Input (π‘₯)Output (5π‘₯+3)
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 (π‘₯)Output (…)
1 9
4 12
10 18

Answer

We have here a function table showing three ordered pairs: (1,9), (4,12), and (10,18). We can notice here when the input increases by 3, from 1 to 4, the output increases also by 3, from 9 to 12. And when it increases by 6, from 4 to 10, the output equally increases by 6 (from 12 to 18). The other remarkable fact is that the difference between the output and the input is the same for the three pairs: it is 8. Both facts suggest that the output is obtained by simply adding 8 to the input, which can be easily checked. Hence, the function rule is π‘₯+8.

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