GCSE Maths

Numbers
32 Topics | 2 Quizzes
Integers
Integers Quiz
Prime Numbers and Composite Numbers
Positive and Negative Numbers
Order of Operations
Square Numbers and Cube Numbers
The Laws of Indices
Square Roots and Cube Roots
Triangular Numbers
The Fibonacci Sequence
Factors
Multiples
The Highest Common Factor
Fractions
Adding and Subtracting Fractions
Multiplication of Fractions
Division of Fractions
Fractional Indices
Decimal Places
Fractions and Decimals
Expressing Fractions as Decimal Numbers
1 of 2
Algebra
29 Topics | 1 Quiz
Solving Inequalities
Graphing Inequalities
Quadratic Inequalities
Linear Simultaneous Equations
Non-linear Simultaneous Equations
Algebraic Equivalence and Proof
Functions
Function Input-Output Tables
Breaking Functions into Step-by-Step Processes
Algebra
2 of 2
Ratio, Proportion and Rates of Change
7 Topics
Ratios
Scaling Ratios
Direct Proportion
Inverse Proportion
Reverse Percentages
Units and Conversions
Scale Factors
Geometry and Measures
16 Topics | 3 Quizzes
2D Shapes
2D Shapes Quiz
3D Shapes
3D Shapes Quiz
Types of angles
Angle Properties
Angle in Parallel Lines
Triangles
Triangles Quiz
Angles in Triangles
Polygons
Lines of Symmetry
Bearings
Loci and constructions
Constructing Triangles
Transformations
Enlargement
Plans and elevations
Areas and Volumes in Map Scale Problems
Probability
10 Topics
Introduction to Probability
Relative Frequency
Sample Space
Probability of Single Events
Probability of Combined Events
Probability Trees
Conditional probability
Set Notation
Venn diagrams
The Complement of a Set
Statistics
2 Topics
Population and samples
Representing data
Vectors
Mensuration and Calculation
7 Topics
Area
Volume
Congruence and similarity
Pythagoras’ theorem
Sine and cosine rule
Area of a triangle
Standard units of measure
Previous Topic
Next Lesson

Breaking Functions into Step-by-Step Processes

GCSE Maths Algebra 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
Previous Topic
Back to Lesson
Next Lesson