All revision notes topics

Formulae and FunctionsEdexcel GCSE Maths: Revision notes

Section 1

How do you substitute numerical values into formulae and expressions?

Substitution means replacing variables (letters) with given numerical values. This is a fundamental skill that underpins all work with formulae.

Method:

  1. Identify the variable(s) you need to replace
  2. Write out the formula clearly
  3. Replace each letter with its value in brackets to avoid sign errors
  4. Follow the order of operations (BIDMAS/BODMAS)
  5. Calculate step-by-step

Example: If P = 2l + 2w (perimeter of a rectangle), and l = 5, w = 3: P = 2(5) + 2(3) = 10 + 6 = 16

When substituting negative numbers, always use brackets: if x = −2 and the formula contains x², write (−2)² = 4, not −2² = −4.

Common situations:

  • Single substitution into linear expressions: straightforward replacement
  • Multiple variables: substitute all at once, not one at a time
  • Powers and roots: ensure correct order of operations
  • Fractions in formulae: keep the fraction structure until the end
Key termssubstitutionvariableexpressionformula
Exam tip

The examiner wants to see clear working showing the substitution step and each calculation stage. Write the substituted values in brackets to demonstrate accuracy.

Common mistake

Students often forget to apply the power to negative numbers correctly: writing −3² as −9 instead of (−3)² = 9. Always bracket negative numbers before squaring.

Section 2

What does rearranging a formula mean and how do you change the subject?

Rearranging (or changing the subject) means using inverse operations to isolate a different variable. This is essential for solving real-world problems where you need a formula expressed in a particular form.

Basic principle: Whatever operation you perform on one side of the equation, you must perform on the other side.

Step-by-step method:

  1. Identify which variable you want to make the subject
  2. Perform inverse operations to isolate it
  3. Addition ↔ Subtraction; Multiplication ↔ Division; Squaring ↔ Square root
  4. Work systematically, dealing with operations in reverse BIDMAS order

Example (simple): Make x the subject of y = 3x + 2

  • Subtract 2: y − 2 = 3x
  • Divide by 3: x = (y − 2)/3

Example (subject appears twice - Higher Tier): Make x the subject of y = 2x + 5x − 3

  • Collect terms with x: y = 7x − 3
  • Add 3: y + 3 = 7x
  • Divide by 7: x = (y + 3)/7

When the subject appears twice with no common factor: You may need to use the quadratic formula or rearrange into a quadratic equation and factorise.

Example: Make p the subject of 3p² + 2p = 15

  • Rearrange: 3p² + 2p − 15 = 0
  • Use the quadratic formula or factorise to find p
Key termsrearrangingsubjectinverse operationscollect like terms
Exam tip

Always show each step of rearrangement clearly. The examiner awards marks for method, not just the final answer. When the subject appears twice, collect all terms containing it on one side first.

Common mistake

Students often reverse the direction of inequality signs incorrectly or forget to apply operations to both sides. Remember: whatever you do to one side, do to the other.

Think of it like this

Think of a formula as a balanced scales: to keep it balanced, any operation on one side must happen on the other side too.

Section 3

What is function notation and how do you work with composite functions? (Higher Tier)

Function notation is a way of expressing relationships between variables using f(x) to mean 'the function f applied to x'.

Key understanding:

  • f(x) is NOT f multiplied by x; it is 'f of x' or 'the output when x is the input'
  • f(2) means substitute x = 2 into the function
  • The letter in brackets is the input; the result is the output

Example: If f(x) = 2x + 3, then:

  • f(2) = 2(2) + 3 = 7
  • f(−1) = 2(−1) + 3 = 1
  • f(x + 1) = 2(x + 1) + 3 = 2x + 5

Composite functions involve applying one function followed by another, written as fg(x) or f(g(x)). This means 'apply g first, then apply f to the result'.

Method for fg(x):

  1. Find g(x) first
  2. Substitute the result into f

Example: If f(x) = x + 2 and g(x) = 3x, find fg(x):

  • Start with g(x) = 3x
  • Apply f: f(g(x)) = f(3x) = (3x) + 2 = 3x + 2
  • So fg(x) = 3x + 2

Finding a specific value: If fg(2) is required, either evaluate g(2) then apply f, or substitute 2 into fg(x).

Order matters: fg(x) ≠ gf(x) in general. Composite functions are not commutative.

Key termsfunction notationcomposite functioninputoutput
Exam tip

For composite functions, examiners expect to see clear working showing g(x) first, then f applied to it. Label each stage clearly and show the substitution step.

Example

If f(x) = x² and g(x) = x − 1, find gf(3): First f(3) = 3² = 9. Then g(9) = 9 − 1 = 8. So gf(3) = 8.

Section 4

How do you find and use inverse functions? (Higher Tier)

An inverse function reverses the effect of the original function. If f(x) gives an output, f⁻¹(x) takes that output and returns the original input. The notation f⁻¹ does NOT mean 1/f.

Key property: f(f⁻¹(x)) = x and f⁻¹(f(x)) = x

Method to find f⁻¹(x):

  1. Write y = f(x)
  2. Rearrange to make x the subject
  3. Swap x and y
  4. The new expression is f⁻¹(x)

Example: Find the inverse of f(x) = 3x + 2

  • Write y = 3x + 2
  • Rearrange for x: 3x = y − 2, so x = (y − 2)/3
  • Swap x and y: y = (x − 2)/3
  • Therefore f⁻¹(x) = (x − 2)/3

Verification: f(f⁻¹(x)) should equal x

  • f(f⁻¹(x)) = f((x − 2)/3) = 3((x − 2)/3) + 2 = (x − 2) + 2 = x ✓

Important considerations:

  • Not all functions have inverses (the function must be one-to-one or injective)
  • For functions to have an inverse, each output must come from exactly one input
  • On a graph, if a horizontal line intersects the curve more than once, no inverse exists
  • The domain of f becomes the range of f⁻¹, and vice versa

Using inverse functions: f⁻¹(k) solves the equation f(x) = k

Key termsinverse functionone-to-one functiondomainrange
Exam tip

Examiners look for clear rearrangement steps when finding inverse functions. Always verify your answer by checking that f(f⁻¹(x)) = x or by substituting a value.

Common mistake

Students often forget to swap x and y at the final step, or they confuse f⁻¹(x) with 1/f(x). Remember: f⁻¹ is the inverse function, not a reciprocal.

Section 5

How do you use iteration to approximate solutions to equations? (Higher Tier)

Iteration is a method of finding approximate solutions by repeatedly applying a formula, getting closer to the true answer each time. It is useful when equations cannot be solved algebraically.

Key idea: Rearrange the equation into the form x = g(x), then repeatedly substitute to generate a sequence that converges to the solution.

Method:

  1. Rearrange the equation into x = g(x) form
  2. Choose a starting value x₀ (often given in the question)
  3. Apply the iterative formula repeatedly: xₙ₊₁ = g(xₙ)
  4. Continue until the value stabilises (successive values agree to the required decimal places)
  5. The stabilised value is the approximate solution

Example: Solve x² + 2x = 5 approximately, starting with x₀ = 1:

  • Rearrange: x² + 2x − 5 = 0, or x = (5 − x²)/2 (so g(x) = (5 − x²)/2)
  • x₀ = 1
  • x₁ = (5 − 1²)/2 = 4/2 = 2
  • x₂ = (5 − 2²)/2 = 1/2 = 1
  • x₃ = (5 − 1²)/2 = 2
  • The sequence oscillates; try another rearrangement

Alternative rearrangement: x = √(5 − 2x) starting with x₀ = 1:

  • x₁ = √(5 − 2) = √3 ≈ 1.732
  • x₂ = √(5 − 2(1.732)) ≈ 1.213
  • x₃ ≈ 1.533, and so on until convergence

Important points:

  • Different rearrangements converge at different rates; some may not converge at all
  • Check if values are stabilising (repeating) to the required accuracy
  • Show at least 3–4 iterations with sufficient decimal places
  • If asked for a solution to d.p. or s.f., continue until you have that precision
Key termsiterationiterative formulaconvergencesuccessive approximation
Exam tip

Examiners expect to see working for each iteration with sufficient decimal places (usually 3–4 d.p.). State clearly when values have stabilised and round to the required precision at the end.

Common mistake

Students often stop iterating too early, before values have converged, or they use rearrangements that oscillate rather than converge. Test your rearrangement with the starting value to check it converges.

Must Know

  • Substitution: Replace letters with numbers in brackets, follow BIDMAS, and use brackets around negative numbers before powers (e.g., (−2)² = 4, not −4)
  • Rearranging formulae: Use inverse operations on both sides; when the subject appears twice, collect like terms together first
  • Function notation: f(x) means 'apply function f to input x'; composite functions fg(x) mean 'apply g first, then apply f to the result'; order matters
  • Inverse functions: Find f⁻¹(x) by writing y = f(x), rearranging for x, then swapping x and y; verify using f(f⁻¹(x)) = x
  • Iteration: Rearrange the equation to x = g(x), apply xₙ₊₁ = g(xₙ) repeatedly with clear working, and stop when successive values stabilise to the required precision
  • Exam technique: Show all working step-by-step, use clear notation, and always verify answers where possible (especially for inverse functions and iterations)

That's the notes covered.

Carry on to the next subtopic.