Functions Notation, Inputs & Outputs — AQA GCSE Mathematics

SPEC A7 • ALGEBRA • FOUNDATION + HIGHER

Functions — Notation, Inputs & Outputs

A function takes an input, does something to it, and returns an output. Learn to read f(x) notation, evaluate functions at any value, and solve for unknown inputs.

What is a Function?

A function is a rule that takes an input, does something to it, and gives you an output. Every input gets exactly one output.

Function

A function is a rule that assigns a unique output to each input. You can think of it as a processing machine: put a number in, the machine transforms it, and a number comes out.

Everyday Examples of Functions

"Multiply by 3 then add 2" — input 5, output 17

"Square it" — input 4, output 16

Temperature conversion — input °C, output °F

Cost calculation — input number of items, output total price

Spec A7 — What's Covered

Foundation covers the concept of input/output functions (often via number machines). Higher adds formal f(x) notation. This lesson covers both.

Number Machines

Number machines are the Foundation-friendly way to represent functions. Each box performs one operation on the number passing through.

Reading a Number Machine

This machine does: input → × 3 → + 2 → output. Each box performs one operation in sequence.

Trace it with input 5:

Start: 5

After × 3: 5 × 3 = 15 (the intermediate value)

After + 2: 15 + 2 = 17 (the final output)

So input 5 → output 17, and in general, input x → output 3x + 2.

AQA Past Paper Pattern

Nov 24 Paper 1 Q2 asked students to complete blank boxes in a number machine so that the overall function matches a given equation like y = 4x + 5. Work through one box at a time.

Introducing f(x) Notation

Number machines are great for beginners. But at Higher tier, we need a more compact way to describe functions. That's where f(x) notation comes in.

Function Notation

f(x) means "the function f evaluated at input x." The letter f is the name of the function; x is the input.

How to Read f(x)

"f of x" — never "f times x"

The brackets tell you x is the input, not a quantity multiplied by f

To evaluate f at a specific value (say x = 5), write f(5) and substitute 5 everywhere x appears in the formula

Any Letter, Same Rule

Functions don't have to be called f. You might see g(x), h(x), p(x) — any letter name works, and the rule is identical: g(a) means substitute a for x in the formula for g. Questions often define two functions at once — AQA Jun 24 P3 Q19 defines both f(x) = x − 3 AND g(x) = 4x − 7 and asks you to work with each.

f(x) Is NOT Multiplication

This is the single most important distinction in function notation. Understanding it correctly prevents a whole family of errors.

❌ Wrong Reading

f(x) = "f times x"

So f(5) = 5f?

And f(2 + 3) = f(2) + f(3)?

All of this is nonsense.

✅ Correct Reading

f(x) = "f evaluated at x"

f(5) means: substitute 5 wherever x appears in the formula

f(2 + 3) = f(5) — work out the bracket first, then apply f

Examiner Alert

What students often write: "f(6)/f(2) = f(6/2) = f(3)" — treating f like it distributes over division.

Why it's wrong: f(a)/f(b) is not generally equal to f(a/b). The brackets around x tell you it's the input to f, not a multiplier you can cancel.

We'll see this exact mistake in Worked Example 4 — a real AQA past paper question (Jun 23 P1 Q13(b)).

Worked Example 1 — Simple Evaluation

01Evaluate the function [2 marks]

Given f(x) = 3x + 2, find f(5).

M — Method. Substitute x = 5 into the formula. Every x becomes 5.

C — Calculate.

f(5) = 3(5) + 2

← M1: correct substitution

= 15 + 2

← multiply first (BIDMAS)

= 17

← A1: final answer

E — Evaluate.f(5) = 17.

Worked Example 2 — Negative Input

When the input is negative, the rules from Lesson 2.06 apply: brackets around the negative, always.

02Evaluate at a negative [2 marks]

Given f(x) = 2x² − 3x, find f(−2).

M — Method. Substitute x = −2, keeping the negative in brackets. Square first (BIDMAS), then work through.

C — Calculate.

f(−2) = 2(−2)² − 3(−2)

← M1: brackets on the negative

= 2(4) − (−6)

← (−2)² = 4, so 2×4 = 8

= 8 + 6

← subtract a negative = add

= 14

← A1: final answer

E — Evaluate.f(−2) = 14.

The Two Sign Flips

Two things to watch: (−2)² = +4 (negative squared is positive), and −3 × (−2) = +6 (negative × negative is positive). Miss either and you'll get a sign error in the final answer.

Worked Example 3 — Finding the Input

Sometimes you're given the output and asked to find the input. Set the function equal to the given value and solve for x.

03Solve for the input [2 marks]

Given f(x) = 3x + 2, solve f(x) = 20.

M — Method. Set the formula equal to 20 and solve the resulting linear equation.

C — Calculate.

3x + 2 = 20

← M1: set up equation

3x = 18

← subtract 2

x = 6

← A1: divide by 3

E — Evaluate.x = 6. Check: f(6) = 3(6) + 2 = 20 ✓

Two Directions

Given input, find output: "Find f(5)" — substitute and compute.

Given output, find input: "Solve f(x) = 20" — set equal, solve for x.

Special Case — f(x) = 0

Solving f(x) = 0 finds the "roots" or "zeros" of the function — the inputs that produce zero output. Graphically, these are the x-values where the graph of y = f(x) crosses the x-axis. You'll use this extensively in Lesson 2.13 (Quadratic Graphs) and Lesson 2.21 (Solving Quadratics by Factorising).

Evaluating at Multiple Inputs

Often a question asks for several outputs from the same function. Build a table — one column for x, one for f(x). This keeps your working organised and the method marks clear.

Example: f(x) = x² − 4x + 5

Input xWorkingOutput f(x)0(0)² − 4(0) + 551(1)² − 4(1) + 5 = 1 − 4 + 522(2)² − 4(2) + 5 = 4 − 8 + 513(3)² − 4(3) + 5 = 9 − 12 + 52−1(−1)² − 4(−1) + 5 = 1 + 4 + 510

Why the Table Helps

Each row is independent — a mistake on one row doesn't carry over. This structure also connects directly to graph plotting (Lesson 2.13): the (x, f(x)) pairs are the coordinates on the graph of y = f(x).

Worked Example 4 — Kai's Claim (Past Paper)

Based on AQA Jun 23 Paper 1, Q13(b). This is a classic question on reading function notation correctly.

04Evaluate and compare [2 marks]

f(x) = kx² where k is a constant. Kai says that f(6)/f(2) is equal to f(3) because 6/2 = 3. Is he correct? Show working to support your answer.

M — Method. Compute f(6), f(2), and f(3) using general k. Then compare f(6)/f(2) with f(3).

C — Calculate.

f(6) = k × 6² = 36k

← substitute x = 6

f(2) = k × 2² = 4k

← substitute x = 2

f(6) / f(2) = 36k / 4k = 9

← M1: the k's cancel

f(3) = k × 3² = 9k

← A1: f(3) depends on k

E — Evaluate.Kai is wrong. f(6)/f(2) = 9 (a constant, independent of k). But f(3) = 9k (depends on k). These are equal only when k = 1, which isn't stated.

Examiner Report

The Jun 23 examiner report on this question flagged that "many students did not show understanding" — most failed to work in terms of the variable k. When proving a general rule, you must keep the constant as a variable. Don't substitute k = 1 or ignore k — that breaks the proof.

Piecewise Functions

Some functions have different rules for different ranges of x. The rule that applies depends on which range your input falls into.

Piecewise Function

A piecewise function is defined by multiple formulas, each valid on a specific range of x values. To evaluate f(a), first check which range a belongs to, then apply the corresponding formula.

Example:

f(x) = 2x + 1 if x < 3f(x) = x² if x ≥ 3

Find f(2). Since 2 < 3, use the first rule: f(2) = 2(2) + 1 = 5.

Find f(3). Since 3 ≥ 3, use the second rule: f(3) = 3² = 9.

Find f(5). Since 5 ≥ 3, use the second rule: f(5) = 5² = 25.

Watch the Inequality Signs

At the boundary value (here, x = 3), read the inequality carefully. x < 3 (strict) means 3 is not included; x ≥ 3 includes 3. Applying the wrong rule at the boundary is a common error.

Weak vs. Strong Reading of f(x)

Same function, same question. Different interpretations lead to very different answers.

❌ Weak Student

f(x) = 2x + 3

Find f(−4):

"f(−4) means f × (−4)"

"= −4f"

"but what is f?"

Marks: 0 / 2

Reads f(x) as "f times x". Treats f as an unknown variable. Cannot proceed.

✅ Strong Student

f(x) = 2x + 3

Find f(−4):

f(−4) = 2(−4) + 3= −8 + 3 = −5

Marks: 2 / 2

Reads f(−4) as "substitute −4 for x." Brackets kept. Arithmetic correct.

The Golden Rule

f(a) = substitute a for x. That's all there is to it.

Common Errors to Avoid

Students Often WriteCorrect VersionWhy It Matters

f(x) read as f × xf(x) = the function f evaluated at xChanges the whole meaning

f(−2) without brackets: −2² = −4f(−2) = … with (−2)² = +4Sign error on squared term

f(6)/f(2) = f(3)f(6)/f(2) = 9, but f(3) = 9k — not equalf is not distributive over division

f(2 + 3) = f(2) + f(3)f(2 + 3) = f(5) — evaluate the bracket firstf is not distributive over addition

Substituting k = 1 when k is a variableKeep k as a symbol and simplify at the endJun 23 P1 Q13(b) examiner-flagged error

Wrong piecewise branch at boundaryRead inequality signs carefully — x < 3 vs x ≥ 3Wrong branch = wrong answer

Forgetting BIDMAS after substitutingEvaluate in BIDMAS order: brackets, powers, ×÷, +−Wrong order = wrong answer

Examiner Alerts — The Big Ones

f(x) Misread as Multiplication

What students write: They read f(−2) as "f times −2" and get confused, or they treat f(a)/f(b) as a fraction that distributes.

Why it loses marks: the function notation is a name (f) applied to an input (the value inside the brackets). It's not arithmetic.

What to write instead: "f of a means: substitute a for x in the formula for f(x)." That's the only operation happening.

Substituting Numbers for Unknown Constants

What students write: On Jun 23 P1 Q13(b), some students silently assumed k = 1 and got apparent agreement.

Why it loses marks: the question says k is a constant — you must keep k symbolic. Substituting a numerical value hides the fact that the claim fails in general.

What to write instead: carry k through every calculation. The working f(6) = 36k, f(2) = 4k, f(3) = 9k makes it obvious that f(6)/f(2) = 9 (constant) but f(3) = 9k (variable).

Sign Errors on Negative Substitutions

What students write: f(−3) when f(x) = x² becomes −9 (wrong) instead of 9 (right).

What to write instead: always wrap the negative input in brackets. (−3)² = (−3) × (−3) = +9.

Exam Technique

Top Exam Tip

When you see f(anything), write "f(anything) = " then substitute "anything" for every x in the formula. That mechanical rule prevents every f(x) notation error on the paper.

Checklist for Every f(x) Question

Did I read f(a) as "f of a", not "f times a"?

Did I put the input in brackets? Especially for negatives.

Did I substitute everywhere x appears? Not just once.

Did I apply BIDMAS to the resulting expression?

For "is it equal?" questions, did I work symbolically with any unknown constants?

Connections to Other Lessons

Lesson 2.06 (Substitution) — same mechanics, just framed with f(x) notation.

Lesson 2.10 (Inverse & Composite) — Higher-only extension of function notation.

Lesson 2.13 (Quadratic Graphs) — the (x, f(x)) table is how you plot y = f(x).

Practice Question

Function evaluation question

The function f is defined by:

f(x) = x² − 5x + 3

(a) Find f(−3). [2 marks]

(b) Solve f(x) = 3. [3 marks]

(a)M

Substitute x = −3, keeping the negative in brackets.

(a)C

f(−3) = (−3)² − 5(−3) + 3

(a)C

= 9 − (−15) + 3 = 9 + 15 + 3

(a)E

f(−3) = 27

(b)M

Set f(x) = 3 and solve the resulting equation.

(b)C

x² − 5x + 3 = 3 → x² − 5x = 0

(b)C

Factorise: x(x − 5) = 0

(b)E

x = 0 or x = 5 — both values give f(x) = 3.

Quick-Fire Checklist

Before you move on, you should be able to do all of these without pausing:

Can You...

Complete a number machine to match a given function rule?

Explain why f(x) is not "f times x"?

Evaluate f(4) when f(x) = 5x − 3? (Answer: 17)

Evaluate f(−2) when f(x) = 3x² + 4x? (Answer: 4)

Solve f(x) = 10 when f(x) = 2x + 4? (Answer: x = 3)

Handle a piecewise function correctly at the boundary?

Work symbolically with an unknown constant (not substitute a specific value)?

One-Line Takeaway

f(x) is a name. To evaluate f(a), substitute a for every x in the formula. That's the only mechanical rule. Everything else — negatives in brackets, BIDMAS, keeping unknown constants symbolic — follows from careful substitution.

Summary

Key Points

Function — a rule that takes an input and returns a unique output.

Number machines — Foundation-friendly diagram: input → operation boxes → output.

f(x) notation — "f of x", meaning the function f evaluated at x. Not multiplication.

Evaluating f(a) — substitute a for every x in the formula; apply BIDMAS.

Negative inputs — always wrap in brackets: f(−2) = … (−2) …

Solving f(x) = k — set the formula equal to k and solve for x.

Piecewise functions — check which range the input falls into, then apply the matching formula.

Unknown constants — keep symbols like k in the working; don't silently substitute k = 1.

Exam Tip

When AQA asks an "is this claim correct?" question about function values (like Kai's claim), always work with any unknown constants symbolically. The Jun 23 examiner report specifically called out students who substituted numerical values — their "proofs" were no proofs at all.