Breaking Functions into Step-by-Step Processes

A function is a mathematical rule that relates input values to output values. The input values are often donated as ‘x‘ and the output values as ‘f(x)‘.

The function rule describes the series of operations that must be performed on the input value to obtain the corresponding output value.

Functions can be broken down into separate operations to better understand the step-by-step process. Each operation represents a step in the process, and the input value passes through each step.

For example, the function f(x) = 2(x + 4) can be broken down into two steps:

Step 1: Add 4 to the input value

(x + 4)

Step 2: Multiply the result from Step 1 by 2

2(x + 4)

Working with Functions in Steps

To understand how a step-by-step function processes input values and generates output values, let’s use the function f(x) = 2(x + 4) with an input value of x = 3:

Step 1: Add 4 to the input value (x + 4):

3 + 4 = 7

Step 2: Multiply the result from Step 1 by 2:

2(x + 4)

2 \times 7 = 14

After passing through both steps, the output value is f(3) = 14.

Representing the Different Steps in a Table

The steps involved in functions can be represented in tables by splitting them into different columns. Each row in the table corresponds to an input value and its corresponding output value, with the intermediate steps displayed in separate columns.

Using the function f(x) = 2(x + 4) as an example, a table representation would look like this:

Input (x)Step 1: (x + 4)Step 2: (2(x + 4))Output (f(x))
11 + 4 = 52 \times 5 = 1010
22 + 4 = 62 \times 6 = 1212
33 + 4 = 72 \times 7 = 1414
44 + 4 = 82 \times 8 = 1616
55 + 4 = 92 \times 9 = 1818

Let’s look at some more examples:

Examples

Example 1:

Function f(x) = (x - 2)^2 + 5

This function can be broken down into three steps:

Step 1: Subtract 2 from the input value:

(x - 2)

Step 2: Square the result from Step 1:

(x - 2)^2

Step 3: Add 5 to the result from Step 2:

(x - 2)^2 + 5

Table representation:

Input (x)Step 1: (x – 2)Step 2: ((x – 2)²)Step 3: ((x – 2)² + 5)Output (f(x))
11 - 2 = -1(-1)^2 = 11 + 5 = 66
22 - 2 = 0(0)^2 = 00 + 5 = 55
33 - 2 = 1(1)^2 = 11 + 5 = 66
44 - 2 = 2(2)^2 = 44 + 5 = 99
55 - 2 = 3(3)^2 = 99 + 5 = 1414

Example 2:

Function g(x) = (3 - (\frac{x}{2}))

This function can be broken down into two steps:

Step 1: Divide the input value by 2:

\frac{x}{2}

Step 2: Subtract the result from Step 1 from 3:

3 - (\frac{x}{2})

Table representation:

Input (x)Step 1: (x / 2)Step 2: (3 – (x / 2))Output (g(x))
1\frac{1}{2} = 0.53 - (\frac{1}{2}) = 2.52.5
2\frac{2}{2} = 13 - (\frac{2}{2}) = 22
3\frac{3}{2} = 1.53 - (\frac{3}{2}) = 1.51.5
4\frac{4}{2} = 23 - (\frac{4}{2}) = 11
5\frac{5}{2} = 2.53 - (\frac{5}{2}) = 0.50.5

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