Function Input-Output Tables

Function input-output tables are a useful tool for visually representing the relationship between inputs and outputs in a function. These tables can help us understand how a function behaves and how changes in the input values affect the output values.

Organising input-output pairs in a table allows us to analyse the function’s behavior and identify patterns that may not be immediately apparent from the function rule.

Creating Input-Output Tables

When creating an input-output table for a function, follow these steps:

Step 1: Begin by selecting input values, which can be a list of specific numbers or a range of values.

Step 2: Apply the function rule to each input value to find the corresponding output values, or the range of the function.

Step 3: Organise the input-output pairs in a table, with one column for input values and another column for output values.

For example, let’s create an input-output table for the linear function f(x) = 2x + 3. We’ll use input values from −2 to 2:

1. Input values: −2, −1, 0, 1, 2

2. Apply the function rule:

f(-2) = 2(-2) + 3 = -1

f(-1) = 2(-1) + 3 = 1

f(0) = 2(0) + 3 = 3

f(1) = 2(1) + 3 = 5

f(2) = 2(2) + 3 = 7

3. Organise the input-output pairs in a table:

xf(x)
21
11
03
15
27

Interpreting Input-Output Tables

Once the input-output table is complete, we can analyse the relationship between input and output values. In our example, we can see that as the input value increases by 1, the output value increases by 2. This shows the linear nature of the function, with a slope of 2.

We can also use input-output tables to make predictions or find missing values. For example, we can predict that when x = 3, the output value f(x) will be 9, following the same pattern.

Examples

Example 1:

For the function f(x) = x^2 - 2x, let’s create an input-output table for the input values -1, 0, 1, 2 and 3.

1. Input values: 1, 0, 1, 2, 3

2. Apply the function to each input value:

f(-1) = (-1)^2 - 2(-1) = 1 + 2 = 3

f(0) = (0)^2 - 2(0) = 0

f(1) = (1)^2 - 2(1) = 1 - 2 = -1

f(2) = (2)^2 - 2(2) = 4 - 4 = 0

f(3) = (3)^2 - 2(3) = 9 - 6 = 3

3. Organise the input-output pairs in a table:

xf(x)
13
00
11
20
33

Example 2:

For the function g(x) = 3 - x, let’s create an input-output table for the input values 2, 4, 6, 8 and 10.

1. Input values: 2, 4, 6, 8, 10

2. Apply the function to each input value:

g(2) = 3 - 2 = 1

g(4) = 3 - 4 = -1

g(6) = 3 - 6 = -3

g(8) = 3 - 8 = -5

g(10) = 3 - 10 = -7

3. Organise the input-output pairs in a table:

xg(x)
21
41
63
85
107

Example 3:

For the function f(x) = x^2 + 1, let’s create an input-output table for the input values 2, 3, 4 and 5.

1. Input values: 2, 3, 4, 5

2. Apply the function to each input value:

f(2) = (2)^2 + 1 = 4 + 1 = 5

f(3) = (3)^2 + 1 = 9 + 1 = 10

f(4) = (4)^2 + 1 = 16 + 1 = 17

f(5) = (5)^2 + 1 = 25 + 1 = 26

3. Organise the input-output pairs in a table:

xf(x)
25
310
417
526

Suppose we want to find the output value for the input x = 4. We can look at the table and see that when x = 4, f(x) = 17. Therefore, for the function f(x) = x^2 + 1, the output value is 17 when the input value is 4.

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