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
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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
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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
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Functions

GCSE Maths Algebra Functions

A function is a mathematical concept that describes the relationship between two sets of numbers, known as the input and output.

In a function, each input value is associated with exactly one output value. Functions help us understand the relationship between variables.

Function Notation

Function notation is a shorter way of expressing a function and its input-output relationship. A function is typically denoted by f(x), where x represents the input value, and f(x) represents the corresponding output value.

For example, let’s look at the function f(x) = 2x + 1. When the input is x = 3, the output is f(3) = 2(3) + 1 = 7.

To evaluate a function for a specific input value, substitute the input value into the function rule and simplify the expression. For example, let’s evaluate the function f(x) = x^2 - 4x + 3 at x = 2:

f(2) = (2)^2 - 4(2) + 3

= 4 - 8 + 3 = -1

So, f(2) = -1

Inverse Functions

An inverse function reverses the input-output relationship of the original function. If the function f takes an input x and produces an output y, then its inverse function, denoted as f^{-1}(x), takes the input y and produces the output x.

To find the inverse of a function, follow these steps:

Step 1: Replace the function notation f(x) with the variable y.

Step 2: Swap the input (x) and output (y) values in the equation.

Step 3: Solve the equation for the new output variable (y).

For example, let’s find the inverse of the function f(x) = 2x + 1:

1. Replace f(x) with y:

y = 2x + 1

2. Swap the input (x) and output (y) values:

x = 2y + 1

3. Solve for the new output variable (y):

x - 1 = 2y

y = \frac{(x - 1)}{2}

So, the inverse function is f^{-1}(x) = \frac{(x - 1)}{2}.

Composite Functions

A composite function is the result of applying one function to the output of another function. If we have two functions, f(x) and g(x), the composite function denoted as (f \: g)(x) or f(g(x)) represents applying the function g first and then applying the function f to the result.

To find a composite function, follow these steps:

Step 1: Identify the functions f(x) and g(x).

Step 2: Write the composite function notation (fg)(x) or f(g(x)).

Step 3: Substitute g(x) into the function f.

For example, let’s find the composite function (fg)(x) for f(x) = 2x + 1 and g(x) = x^2:

1. Identify the functions:

f(x) = 2x + 1

g(x) = x^2

2. Write the composite function notation:

(fg)(x) = f(g(x))

3. Substitute g(x) into the function f:

(fg)(x) = f(x^2)

= 2(x^2) + 1

So, the composite function (fg)(x) = 2x^2 + 1.

Examples

Example 1:

Given the function f(x) = 3x - 7, find f(4).

To find f(4), substitute x = 4 into the function:

f(4) = 3(4) - 7

= 12 - 7

= 5

So, f(4) = 5.

Example 2:

Find the inverse of the function f(x) = 4x - 5.

To find the inverse function, follow these steps:

1. Replace f(x) with y: y = 4x - 5

2. Swap x and y: x = 4y - 5

3. Solve for y: 4y = x + 5 y = \frac{x + 5}{4}

So, the inverse function is f^{-1}(x) = \frac{x + 5}{4}.

Example 3:

Given the functions f(x) = 2x + 3 and g(x) = x^2 - 1, find (fg)(x) and (gf)(x).

1. Find (fg)(x) = f(g(x)):

Substitute g(x) into the function f: (fg)(x) = f(x^2 - 1)

= 2(x^2 - 1) + 3

= 2x^2 - 2 + 3

= 2x^2 + 1

So, (fg)(x) = 2x^2 + 1.

2. Find (gf)(x) = g(f(x)):

Substitute f(x) into the function g: (gf)(x) = g(2x + 3)

= (2x + 3)^2 - 1

= (4x^2 + 12x + 9) - 1

= 4x^2 + 12x + 8

So, (gf)(x) = 4x^2 + 12x + 8.

Example 4:

Given the function h(x) = \frac{x^2 - 3x + 2}{x - 1}, find h(2).

To find h(2), substitute x = 2 into the function:

h(2) = \frac{(2)^2 - 3(2) + 2}{2 - 1} = \frac{4 - 6 + 2}{1} = \frac{0}{1} = 0

So, h(2) = 0.

Example 5:

Question: Find the inverse of the function p(x) = \frac{3x - 4}{2}.

To find the inverse function, follow these steps:

1. Replace p(x) with y: y = \frac{3x - 4}{2}

2. Swap x and y: x = \frac{3y - 4}{2}

3. Solve for y: 2x = 3y - 4

3y = 2x + 4

y = \frac{2x + 4}{3}

So, the inverse function is p^{-1}(x) = \frac{2x + 4}{3}.

Example 6:

Question: Given the functions m(x) = \sqrt{x} and n(x) = x^2 + 4, find (mn)(x) and (nm)(x).

1. Find (mn)(x) = m(n(x)):

Substitute n(x) into the function m: (mn)(x) = m(x^2 + 4)

= \sqrt{x^2 + 4}

So, (mn)(x) = \sqrt{x^2 + 4}.

2. Find (nm)(x) = n(m(x)):

Substitute m(x) into the function n: (nm)(x) = n(\sqrt{x})

= (\sqrt{x})^2 + 4

= x + 4

So, (nm)(x) = x + 4.

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