IA IAL ATLAS ← Pure Mathematics
Pure Mathematics • Chapter 01

Algebra
& Functions

A deep, structured chapter for Edexcel International A Level Pure Mathematics. From algebraic language and manipulation to quadratics, inequalities, composite functions, inverses, transformations, modelling and exam-level reasoning.

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Algebra & Functions Atlas

Use this map to jump directly to a major concept.

01 — Algebra Foundations 02 — Manipulation & Identities 03 — Algebraic Fractions 04 — Indices 05 — Surds 06 — Quadratics 07 — Simultaneous Equations 08 — Inequalities 09 — Functions 10 — Graphs & Transformations 11 — Modelling & Reasoning 12 — Exam Technique 13 — Formula Atlas 14 — Practice Bank 15 — 100Q Mastery Exam
01 • Foundations

Algebra Foundations

Build the language first. Most difficult algebra errors begin with weak notation or uncontrolled manipulation.

1.1 What is algebra?

Definition: Algebra is a system for representing numbers, quantities and relationships using symbols. A letter such as x can represent an unknown value, a variable quantity or a general number.

A numerical expression contains numbers and operations. An algebraic expression can also contain variables. For example, 7 + 3 × 4 is numerical, while 3x + 7 is algebraic.

The important distinction is that an expression does not contain an equals sign, whereas an equation states that two expressions are equal. An identity is true for every allowed value of its variable.

Expression: 3x + 5
Equation: 3x + 5 = 17
Identity: (x + 2)² ≡ x² + 4x + 4

Key vocabulary

  • Term: a part separated by + or − signs.
  • Coefficient: numerical multiplier of a variable.
  • Constant: a term with no variable.
  • Factor: a quantity multiplied by another quantity.
  • Variable: a symbol representing a value that may vary.
  • Power/index: tells how many times a base is multiplied by itself.
  • Polynomial: an expression made from terms involving non-negative integer powers of variables.
Worked example

Identify the terms, coefficient and constant in:

5x² − 3x + 11

The terms are 5x², −3x and 11.

The coefficient of x² is 5. The coefficient of x is −3. The constant is 11.

Common trap: The coefficient of −x is −1, not 1. The coefficient of x is 1.
Checkpoint: Before manipulating an expression, identify its terms and signs. Treat the sign as part of the term.

1.2 Like terms

Terms can only be collected directly when their variable parts are identical. The powers matter.

3x + 5x = 8x
7x² − 2x² = 5x²
4xy + 3xy = 7xy

But x and x² are not like terms, and xy and x²y are not like terms.

Worked example
4x² + 7x − 3 + 2x² − 5x + 9

Group like terms:

(4x² + 2x²) + (7x − 5x) + (−3 + 9)
= 6x² + 2x + 6

1.3 Substitution

Substitution means replacing a variable by a particular value. Always use brackets when substituting a negative number.

Worked example

Evaluate 2x² − 5x + 3 when x = −2.

2(−2)² − 5(−2) + 3
= 2(4) + 10 + 3
= 21
Common trap: Writing −2² instead of (−2)². By order of operations, −2² = −4, whereas (−2)² = 4.

1.4 Rearranging equations

To solve an equation, perform the same reversible operation on both sides. Think of an equation as a balance.

Worked example
5x − 7 = 18
5x = 25
x = 5

For more complicated equations, remove operations in reverse order. If the original expression was built as:

x → ×5 → −7 → =18

undo it as:

18 → +7 → ÷5 → x=5

1.5 Algebraic language and restrictions

Algebraic simplification can introduce restrictions that are invisible in the final expression. For example:

(6x²)/(3x) = 2x

This cancellation is valid only when x ≠ 0, because the original expression is undefined at zero.

Exam lens: If a denominator contains a variable, check whether cancelling creates or hides a restriction.
02 • Manipulation

Expansion, Factorisation & Identities

The central algebraic toolkit used repeatedly throughout Pure Mathematics.

2.1 Expanding a single bracket

Distributive law: Every term outside a bracket must multiply every term inside it.
a(b+c)=ab+ac
Worked example
3(x+5)=3x+15
Negative multiplier
−4(2x−3)=−8x+12

The negative sign multiplies both terms.

Common trap: Do not multiply only the first term. The bracket is a package: every term must be affected.

2.2 Expanding two brackets

(a+b)(c+d)=ac+ad+bc+bd
Worked example
(x+4)(x+7)
=x²+7x+4x+28
=x²+11x+28
Harder example
(2x−3)(x+5)
=2x²+10x−3x−15
=2x²+7x−15

2.3 Important identities

Square of a sum (a+b)²=a²+2ab+b²
Square of a difference (a−b)²=a²−2ab+b²
Difference of two squares a²−b²=(a−b)(a+b)
Cube of a sum (a+b)³=a³+3a²b+3ab²+b³
Identity in action
(x−6)²=x²−12x+36

An identity is not the same as an equation that is only true for particular values. The symbol ≡ is often used to emphasize that two expressions are identical for every allowed value.

2.4 Factorisation by common factor

Factorisation is the reverse of expansion. First identify the greatest common factor.

Worked example
3x²+12x

The common factor is 3x.

3x(x+4)
Common trap: Always multiply your factorised answer back out to verify it.

2.5 Factorising quadratics

For a monic quadratic:

x²+bx+c

find two numbers whose product is c and whose sum is b.

Worked example
x²+9x+20

Numbers multiplying to 20 and adding to 9 are 4 and 5.

x²+9x+20=(x+4)(x+5)
Negative coefficients
x²−7x+12
=(x−3)(x−4)
Leading coefficient greater than 1
2x²+7x+3
=(2x+1)(x+3)

The cross terms are 6x+x=7x.

2.6 Algebraic proof using identities

A proof must establish the result generally, not merely test a few values.

Proof

Show that the sum of two consecutive integers is odd.

Let the first integer be n.
The next integer is n+1.
Sum = n+(n+1)=2n+1

Since 2n is even, 2n+1 is odd. Therefore the result holds for every integer n.

03 • Fractions

Algebraic Fractions

Factor first, identify restrictions, then simplify with control.

3.1 Simplifying algebraic fractions

Key idea: You may cancel common factors, not arbitrary terms.
Worked example
(6x²y)/(3xy)
=2x

provided x≠0 and y≠0.

Do not do this: (x+3)/x ≠ 3. The x is not a factor of the entire numerator.

3.2 Adding algebraic fractions

Fractions must have a common denominator before their numerators can be combined.

Worked example
1/x + 2/x = 3/x
Different denominators
1/x + 1/2

Common denominator = 2x.

=2/(2x)+x/(2x)
=(x+2)/(2x)

3.3 Algebraic fractions with factorisation

Worked example
(x²−9)/(x²+5x+6)

Factorise both:

(x−3)(x+3) / [(x+2)(x+3)]
=(x−3)/(x+2)

Original restrictions:

x≠−2, −3

Even though x+3 cancels, the original expression was undefined at x=−3.

Exam lens: Write restrictions from the original denominator before cancelling.
04 • Indices

Indices & Powers

Understand why the laws work instead of memorising isolated patterns.

4.1 Positive integer indices

a⁴=a×a×a×a

The index counts repeated multiplication of the same base.

4.2 Multiplication law

a^m × a^n = a^(m+n)
Worked example
2³×2⁴=2⁷=128

The bases must be the same. You add the indices because you are joining repeated factors.

4.3 Division law

a^m/a^n=a^(m−n)
7⁶/7²=7⁴

4.4 Power of a power

(a^m)^n=a^(mn)
(3²)^4=3⁸
Common trap: Do not add 2 and 4 here. The operation is multiplication of powers.

4.5 Zero and negative indices

a⁰=1,   a≠0

A negative power represents a reciprocal:

a^(−n)=1/a^n
5^(−2)=1/25

4.6 Fractional indices

a^(1/n)=ⁿ√a
a^(m/n)=ⁿ√(a^m)
Worked example
16^(1/2)=4
27^(2/3)=(∛27)²=3²=9
Another form
9^(−1/2)=1/√9=1/3

4.7 Algebraic powers

x³×x^(−1)=x²

For nonzero x, subtract the powers:

3+(−1)=2
Domain warning: When fractional powers are involved, the allowed real domain may matter. Do not silently assume every real x is valid.
05 • Surds

Surds & Exact Form

Keep irrational quantities exact rather than converting them prematurely into decimals.

5.1 Simplifying square roots

Look for a square factor inside the radical.

Worked example
√50=√(25×2)=5√2
√72=√(36×2)=6√2

5.2 Adding surds

Only like surds can be collected.

√12+√27
=2√3+3√3
=5√3
Common trap: √2+√3 cannot be simplified to √5.

5.3 Multiplying surds

√a × √b = √(ab)
√3×√12=√36=6

5.4 Rationalising a denominator

An exact answer is often expected with no irrational denominator.

Single surd denominator
1/√5 × √5/√5 =√5/5

5.5 Conjugates and binomial surds

For a denominator such as a+√b, multiply by the conjugate a−√b.

(a+b)(a−b)=a²−b²
Worked example
1/(3+√2)
×(3−√2)/(3−√2)
=(3−√2)/(9−2)
=(3−√2)/7
Exam lens: If the denominator has two terms and one contains a square root, look for a conjugate.
06 • Quadratics

Quadratic Equations & Graphs

Roots, discriminant, completing the square, turning points, parameters and graph interpretation.

6.1 What is a quadratic?

Definition: A quadratic expression has the general form ax²+bx+c, where a≠0.

Its graph is a parabola. The sign of a determines whether it opens upward or downward.

a>0 → opens upward → minimum
a<0 → opens downward → maximum

6.2 Solving by factorisation

Worked example
x²−5x+6=0
(x−2)(x−3)=0

Therefore:

x−2=0 or x−3=0
x=2 or x=3
Zero-product principle: If AB=0, then A=0 or B=0.

6.3 Quadratic formula

x = [−b ± √(b²−4ac)]/(2a)

For ax²+bx+c=0, identify a, b and c carefully before substituting.

Worked example

Solve:

2x²−3x−2=0

Here:

a=2, b=−3, c=−2
x=[3±√(9+16)]/4
x=[3±5]/4
x=2 or x=−1/2
Common trap: If b is negative, the formula contains −(−3). Keep brackets when substituting.

6.4 Discriminant

Δ=b²−4ac

The discriminant tells you the number of real roots:

Δ > 0 Two distinct real roots.
Δ = 0 One repeated real root.
Δ < 0 No real roots.
Geometric meaning It describes how the parabola intersects the x-axis.
Parameter question

For:

x²+6x+k=0

Find k for equal roots.

Δ=6²−4(1)(k)=0
36−4k=0
k=9

6.5 Completing the square

Completing the square rewrites a quadratic in vertex form.

x²+10x+21
=x²+10x+25−4
=(x+5)²−4

This immediately gives the turning point:

(−5,−4)

6.6 Turning points and axis of symmetry

For:

y=a(x−h)²+k

the turning point is (h,k) and the axis of symmetry is x=h.

y=(x−3)²+2

Turning point = (3,2).

Minimum value = 2.

Axis = x=3.

y=(x+4)²−7

Turning point = (−4,−7).

6.7 Roots and coefficients

If:

ax²+bx+c=a(x−α)(x−β)

then:

α+β=−b/a
αβ=c/a
Worked example

For 2x²−10x+12=0:

sum of roots = −(−10)/2 = 5
product = 12/2 = 6

6.8 Quadratic inequalities

Solve the equation first to find the critical points. Then determine which intervals satisfy the inequality.

Worked example
x²−9>0
(x−3)(x+3)>0

Critical points: −3 and 3.

The upward-opening parabola is positive outside its roots:

x<−3 or x>3
x²−4x+3≤0
(x−1)(x−3)≤0
1≤x≤3

6.9 Parameter conditions

Many exam questions ask for a range of a parameter such that a quadratic has: two roots, no roots, a repeated root, roots of a particular sign, or roots in an interval.

Parameter reasoning

For a quadratic to have two distinct real roots, impose:

b²−4ac>0

For exactly one repeated real root:

b²−4ac=0

For no real roots:

b²−4ac<0
Exam lens: Translate the words into a mathematical condition before doing algebra. “Two distinct real roots” means Δ > 0, not Δ ≥ 0.
07 • Systems

Simultaneous Equations

Two equations impose two conditions. The solution is the point satisfying both.

7.1 Linear + linear

Elimination
x+y=7
x−y=1

Add the equations:

2x=8
x=4

Substitute:

4+y=7
y=3

Solution:

(x,y)=(4,3)

7.2 Linear + quadratic

Substitute the linear expression into the quadratic equation. This converts the system into a single-variable quadratic.

Method
y=x+1
y=x²−3x+1

Since both equal y:

x+1=x²−3x+1
x²−4x=0
x(x−4)=0
x=0 or x=4

Then find the corresponding y values.

Exam lens: A quadratic + line can have 0, 1 or 2 intersection points. The algebraic number of real solutions has a geometric interpretation.

7.3 Practice systems

1. x+y=7, x−y=1
Answer: (4,3)
2. 2x+y=9, x+y=6
Answer: (3,3)
3. 3x+2y=12, x+y=5
Answer: (2,3)
4. 4x−y=9, x+y=6
Answer: (3,3)
5. 5x+y=17, x−y=1
Answer: (3,2)
6. 2x+3y=13, x+y=5
Answer: (2,3)
7. 3x−y=5, x+y=7
Answer: (3,4)
8. x+2y=8, 2x−y=1
Answer: (2,3)
9. 2x+y=1, x−y=2
Answer: (1,−1)
10. 3x+y=10, 2x−y=0
Answer: (2,4)
08 • Inequalities

Inequalities & Solution Sets

Understand the direction, interval structure and graphical meaning of inequality solutions.

8.1 Linear inequalities

3x+2>11
3x>9
x>3

8.2 Reversing the sign

When multiplying or dividing an inequality by a negative number, reverse its direction.

−4x>20

Divide by −4 and reverse >:

x<−5
Common trap: The inequality sign changes only when multiplying or dividing by a negative quantity. Adding or subtracting does not reverse it.

8.3 Compound inequalities

2
−1

8.4 Quadratic inequalities using a sign diagram

x²−4x+3≤0
(x−1)(x−3)≤0

Critical values:

x=1,3

Since the parabola opens upward, it is below or on the x-axis between the roots:

1≤x≤3
Checkpoint: For quadratic inequalities, do not simply solve the equation and stop. The question asks for intervals where the expression has the required sign.
09 • Functions

Functions, Domain, Range & Inverses

The function viewpoint turns algebra into a system of inputs, outputs and transformations.

9.1 What is a function?

Function: A function assigns exactly one output to every permitted input.

We can think of a function as a machine:

input x → function f → output f(x)
f(x)=2x+5

Then:

f(3)=2(3)+5=11

9.2 Function notation

The notation f(x) means “the output of function f when the input is x”. It does not mean f multiplied by x.

f(x)=x²−1
f(−2)=(−2)²−1=3

9.3 Finding an input from an output

f(x)=3x−4

Find x when f(x)=11:

3x−4=11
3x=15
x=5

9.4 Domain and range

Domain: the set of allowed inputs.
Range: the set of outputs produced.

Restrictions can come from square roots, denominators, logarithms or the context of a model.

f(x)=√x

For real outputs:

x≥0
f(x)=1/(x−2)

The denominator cannot be zero:

x≠2

9.5 One-to-one functions

A function is one-to-one if different inputs produce different outputs. This is important because an inverse function must itself be a function.

f(x)=x²

On all real numbers:

f(2)=4
f(−2)=4

Therefore x² is not one-to-one on all real numbers. A restriction such as x≥0 makes it one-to-one.

9.6 Composite functions

A composite function means applying one function after another.

(f∘g)(x)=f(g(x))
Worked example
f(x)=x+2
g(x)=3x

Find f(g(4)):

g(4)=12
f(12)=14
f(g(4))=14
Algebraic composition
f(x)=2x+1
g(x)=x−4
f(g(x))=2(x−4)+1
=2x−7
Order matters: Generally f(g(x)) ≠ g(f(x)).

9.7 Inverse functions

An inverse function reverses the original mapping. If:

f: x → y

then:

f⁻¹: y → x
Worked example
f(x)=3x+2

Write:

y=3x+2

Swap x and y:

x=3y+2

Solve for y:

x−2=3y
y=(x−2)/3

Therefore:

f⁻¹(x)=(x−2)/3

9.8 Checking an inverse

A strong verification method is composition:

f(f⁻¹(x))=x

and, on the appropriate domain:

f⁻¹(f(x))=x
f(x)=3x+2
f⁻¹(x)=(x−2)/3
f(f⁻¹(x)) =3[(x−2)/3]+2 =x−2+2 =x

9.9 Restrictions and inverse functions

If a function is not one-to-one on its original domain, its domain may need to be restricted.

f(x)=x²

Restrict to x≥0. Then:

y=x²

Swap:

x=y²

Because the inverse output must be non-negative:

y=√x

So:

f⁻¹(x)=√x
10 • Graphs

Function Graphs & Transformations

Read algebraic changes as geometric movements and changes in shape.

10.1 Translation

Starting with:

y=f(x)

Then:

f(x)+a moves the graph upward by a.
f(x)−a moves the graph downward by a.
f(x−a) moves right by a.
f(x+a) moves left by a.
Most common transformation trap: The horizontal sign appears reversed. f(x−3) moves right 3, not left 3.

10.2 Reflection

y=−f(x) Reflection in the x-axis.
y=f(−x) Reflection in the y-axis.

10.3 Vertical scaling

y=af(x)

If |a|>1, vertical stretching occurs. If 0<|a|<1, vertical compression occurs. If a is negative, there is also reflection in the x-axis.

10.4 Horizontal scaling

y=f(bx)

The horizontal scale factor is:

1/|b|

This is another place where the algebraic multiplier can behave opposite to first intuition.

10.5 Quadratic graph interpretation

y=−x²+4x+1

Because the coefficient of x² is negative, the graph opens downward. Therefore the turning point is a maximum.

Complete the square:

y=−(x²−4x)+1
y=−[(x−2)²−4]+1
y=−(x−2)²+5

Maximum point:

(2,5)

10.6 Intersections and solutions

Solving two equations simultaneously can be interpreted as finding graph intersections. For example:

x²−3x+1 = 2x+4

The solutions are x-coordinates where the quadratic and line meet.

Exam lens: Whenever an algebraic equation is linked to a graph, ask: “What does this solution represent geometrically?”
11 • Modelling

Modelling, Interpretation & Reasoning

Algebra becomes mathematics when symbols are connected to a real or mathematical context.

11.1 Building an algebraic model

A model translates a verbal relationship into mathematics. The key stages are:

  1. Define the variable.
  2. Translate the given relationship.
  3. Build the equation or function.
  4. Solve mathematically.
  5. Check whether the solution makes sense in context.
Example

A rectangle has width x cm and length 3 cm greater than its width.

width=x
length=x+3

Area:

A=x(x+3)=x²+3x

11.2 Rejecting mathematically valid solutions

An equation can produce solutions that are algebraically valid but impossible in context.

If x represents a physical length, then x must normally satisfy:

x>0

A negative algebraic root may therefore be rejected.

Exam lens: Never stop after solving. Return to the original meaning of the variable.

11.3 Exact versus approximate answers

Keep exact values such as √5 or (3+√7)/2 until the final stage unless the question specifically requires a decimal approximation. Premature rounding can alter later calculations.

11.4 Mathematical reasoning

A strong solution does not only state the answer. It establishes why the answer follows. Useful reasoning phrases include:

  • “Since the expression is always non-negative…”
  • “Because the denominator cannot equal zero…”
  • “The discriminant must be positive because two distinct real roots are required…”
  • “The coefficient of x² is negative, so the parabola opens downward…”
  • “The domain is restricted to…”
12 • Exam Technique

Exam Strategy & Error Control

Knowing the mathematics is only half the job. The other half is controlling your method.

12.1 The algebra checklist

  1. Read the command word: simplify, solve, show, prove, find, hence, state.
  2. Copy signs carefully.
  3. Write substitutions with brackets.
  4. Keep exact values where possible.
  5. Check denominators for restrictions.
  6. For quadratics, identify a, b and c before using the formula.
  7. For inequalities, check whether multiplying/dividing by a negative reverses the sign.
  8. For functions, preserve composition order.
  9. For modelling, check the physical meaning.
  10. Substitute answers back into the original equation whenever practical.

12.2 “Show that” questions

A “show that” question gives you a target. You must produce a chain of mathematically valid steps ending at that target. Do not simply write the target as if it were known.

12.3 “Hence” questions

“Hence” usually means the previous result is intended to be used. Look for a substitution, transformation or relationship that makes the previous answer useful.

12.4 Common algebra errors

Sign error
Especially after expanding negative brackets.
Wrong cancellation
Cancelling terms instead of factors.
Index error
Adding powers when multiplying powers is correct, but not for powers of powers.
Quadratic formula
Forgetting the ±.
Discriminant
Confusing Δ=0 with Δ>0.
Inequality
Forgetting to reverse the sign after dividing by a negative.
Functions
Mixing up f(g(x)) and g(f(x)).
Inverse
Forgetting to swap x and y before solving.
Domain
Ignoring forbidden inputs.
Context
Accepting an impossible negative length/time.

12.5 Final verification routine

1. Is the algebra legal?
2. Is the answer in the correct form?
3. Does it satisfy the original equation?
4. Does the domain allow it?
5. Does the context allow it?
13 • Formula Atlas

Algebra & Functions Quick Reference

A compact revision layer after the detailed teaching sections.

Expansion a(b+c)=ab+ac
Square (a+b)²=a²+2ab+b²
Difference squares a²−b²=(a−b)(a+b)
Index multiplication a^m a^n=a^(m+n)
Index division a^m/a^n=a^(m−n)
Power of power (a^m)^n=a^(mn)
Negative index a^(−n)=1/a^n
Fractional index a^(m/n)=ⁿ√(a^m)
Quadratic formula x=[−b±√(b²−4ac)]/(2a)
Discriminant Δ=b²−4ac
Roots sum α+β=−b/a
Roots product αβ=c/a
Composite (f∘g)(x)=f(g(x))
Inverse check f(f⁻¹(x))=x
Up translation f(x)+a
Right translation f(x−a)
14 • Practice Bank

Mixed Practice

Short targeted questions before the full mastery examination.

01. Expand (x+4)(x+7).
x²+11x+28
02. Expand (2x−3)(x+5).
2x²+7x−15
03. Factorise x²+9x+20.
(x+4)(x+5)
04. Factorise 3x²+12x.
3x(x+4)
05. Simplify 4x+7−2x+3.
2x+10
06. Expand (x−6)².
x²−12x+36
07. Factorise 2x²+7x+3.
(2x+1)(x+3)
08. Simplify 6x²y/(3xy).
2x, with original restrictions.
09. Evaluate 2³×2⁴.
2⁷
10. Evaluate 27^(2/3).
9
11. Simplify √50.
5√2
12. Rationalise 1/√5.
√5/5
13. Solve x²−5x+6=0.
x=2,3
14. Find the discriminant of x²−8x+16.
0
15. Complete the square: x²+10x+21.
(x+5)²−4
16. Solve x²−9>0.
x<−3 or x>3
17. Solve 5−2x≤1.
x≥2
18. If f(x)=2x+5, find f(3).
11
19. If f(x)=x²−1, find f(−2).
3
20. Find the inverse of 3x+2.
(x−2)/3
21. If f=x+2 and g=3x, find f(g(4)).
14
22. Domain of √x.
x≥0
23. Domain restriction of 1/(x−2).
x≠2
24. Turning point of (x−3)²+2.
(3,2)
25. Minimum of (x+4)²−7.
−7
26. Does x² have an inverse on all real numbers?
No; it is not one-to-one.
27. Solve 3x+2>11.
x>3
28. Solve −4x>20.
x<−5
29. Solve x²−4x+3≤0.
1≤x≤3
30. Find inverse of 4x−7.
(x+7)/4
15 • Mastery Assessment

IAL ATLAS Mastery Exam

100 compulsory questions • 120 minutes • no skipping ahead • automatic marking

Before you start: This is an independent IAL ATLAS practice assessment, not an official Pearson Edexcel examination. The questions are designed to test the chapter concepts covered above.
100 Questions Every question must be answered.
120 Minutes The timer runs continuously.
No Skip Ahead You must answer the current question before moving forward.
Instant Analysis Score, topic breakdown and wrong-answer explanations appear after submission.