Inverse & Composite Functions

SPEC A7 • ALGEBRA • HIGHER TIER ONLY

Inverse functions undo what a function does. Composite functions chain two functions together. Both appear regularly on Higher papers — and both trip students up on notation.

Two New Ideas Built on f(x)

Lesson 2.09 taught you to evaluate functions. This lesson adds two Higher-only ideas that combine or reverse functions.

Inverse Function f⁻¹(x)

Undoes what f(x) does.

If f(3) = 11, then f⁻¹(11) = 3.

Every input you put through f and then f⁻¹ gets you back where you started.

Composite Function fg(x)

Chains two functions.

fg(x) = f(g(x)): apply g first, then f.

The output of one function becomes the input of the next.

Spec A7 (Higher)

The AQA spec A7 explicitly requires understanding of f(x), fg(x) and f⁻¹(x) notation at Higher tier. Both skills appear on most Higher papers — sometimes as short 2-mark questions, sometimes combined in 4–5 mark problems.

Inverse Functions — What They Do

An inverse function reverses what the original function does. Think of it as a machine that runs backwards.

Inverse Function

The inverse function f⁻¹(x) takes the output of f and returns the original input. Formally: if f(a) = b, then f⁻¹(b) = a.

The Key Identity

f(f⁻¹(x)) = x and f⁻¹(f(x)) = x

Applying f then f⁻¹ (or f⁻¹ then f) gets you back to where you started. This gives you a way to check your inverse.

Finding f⁻¹(x) — The Procedure

There's a reliable 3-step recipe for finding the inverse of any function. Learn the steps and use them every time — do not try to "spot" the inverse.

1Write y = f(x). Replace "f(x)" with y.

2Swap x and y. Every x becomes y and vice versa.

3Solve for y. Rearrange using the techniques from Lesson 2.07. The result is f⁻¹(x).

Do Not Symbol-Flip

Common mistake: "inverse of 3x + 2 is 2x + 3" — flipping the operations visually. This is wrong. You must actually do the algebra: swap x and y, then rearrange. Symbol-flipping gets the wrong answer almost every time.

Worked Example 1 — Linear Inverse

01Find the inverse [2 marks]

f(x) = 3x + 2. Find f⁻¹(x).

M — Method. Write y = f(x), swap x and y, solve for y.

C — Calculate.

y = 3x + 2

← step 1: write y = f(x)

x = 3y + 2

← step 2: swap x ↔ y

x − 2 = 3y

← M1: subtract 2

y = (x − 2)/3

← A1: divide by 3

E — Evaluate.f⁻¹(x) = (x − 2)/3

Check: f(f⁻¹(x)) = 3 × (x − 2)/3 + 2 = (x − 2) + 2 = x ✓

Always Check

After finding an inverse, substitute it back: f(f⁻¹(x)) must simplify to x. If it doesn't, there's an error in your algebra. This 30-second check catches most mistakes.

Why the Procedure Works — The Intuition

f(x) = 3x + 2 performs "×3 then +2" on the input. To undo this, reverse both the operations and their order: "−2 then ÷3". That's exactly (x − 2)/3. The swap-and-solve procedure is a mechanical way to do this undoing automatically — but thinking through "what did f do, and how do I undo it?" is a powerful sanity check.

Composite Functions — Chaining Two Functions

A composite function runs the output of one function into another. You're feeding the result of one machine into the next.

Composite Function

fg(x) means "apply g first, then apply f to the result." In standard notation: fg(x) = f(g(x)).

Critical — Read Right to Left

In the notation fg(x), the function closer to the x — the inner one — is applied first. That's g. Then f is applied to the result. It's the opposite of how you'd read English. This is the most common error on composite questions — see Slide 11.

Worked Example 2 — Numerical Composite (Past Paper)

Based on AQA Jun 24 Paper 3, Q19(a). This is the simplest type of composite question — plug in a number, compute step by step.

02Evaluate fg(6) [2 marks]

f(x) = x − 3 and g(x) = 4x − 7. Work out the value of fg(6).

M — Method. fg(6) means "apply g to 6 first, then apply f to the result." Two steps: compute g(6), then plug that into f.

C — Calculate.

Step 1: g(6) = 4(6) − 7

← M1: inner function first

= 24 − 7 = 17

← the intermediate value

Step 2: f(17) = 17 − 3

← apply f to g(6)

= 14

← A1: final answer

E — Evaluate.fg(6) = 14.

The Two-Step Rule

For numerical composite questions, always compute in two separate lines: (1) find g(input), (2) feed that into f. Never try to do it all in one line — that's where sign errors and order mix-ups creep in.

Worked Example 3 — Algebraic Composite (Multiple Choice)

Based on AQA Jun 22 Paper 3, Q22 — a short 1-mark multiple choice that rewards correctly identifying the inner function.

03Find fg(x) [1 mark]

f(x) = 3x and g(x) = x². Circle the expression for fg(x). Options: 3x² | 9x² | 3x³ | 9x⁴

M — Method. fg(x) means g first, then f. Start with g(x) = x², then feed into f.

C — Calculate.

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

← definition of composite

= f(x²)

← substitute g(x) = x²

= 3(x²)

← apply f: multiply by 3

= 3x²

← B1: correct answer

E — Evaluate.Circle 3x².

Why 9x² Is a Tempting Wrong Answer

Students who read fg left-to-right apply f first: f(x) = 3x, then g squares it: (3x)² = 9x². Wrong. The correct order is g first, then f, giving 3x² — not 9x².

Worked Example 4 — Algebraic Composite (Past Paper)

Based on AQA Jun 23 Paper 3, Q18. This is the standard 3-mark AQA composite question — substitute, expand, simplify.

04Show that fg(x) = … [3 marks]

f(x) = x² + 6x and g(x) = 2x + 4. Show that fg(x) = 4x² + 28x + 40.

M — Method. fg(x) = f(g(x)). Substitute g(x) = 2x + 4 everywhere x appears in f, expand, and simplify.

C — Calculate.

fg(x) = f(2x + 4)

← M1: substitute inner function

= (2x + 4)² + 6(2x + 4)

← replace x in f's formula

= (4x² + 16x + 16) + (12x + 24)

← M1: expand both brackets

= 4x² + 28x + 40

← A1: collect like terms

E — Evaluate.fg(x) = 4x² + 28x + 40, as required. ✓

Part (b) Follow-Up — Solve fg(x) = −5

The real Jun 23 Q18 then asks [3 marks] to solve fg(x) = −5 using the result from part (a):

4x² + 28x + 40 = −5 → 4x² + 28x + 45 = 0 (move −5 across)

Quadratic formula: x = (−28 ± √(784 − 720))/8 = (−28 ± 8)/8

Solutions: x = −2.5 or x = −4.5

Always read the full multi-part question before starting — knowing where you're headed shapes how you present part (a).

Why fg(x) ≠ gf(x)

Composite function order matters enormously. Using the same f and g, switching the order usually gives a completely different function.

Let f(x) = 2x + 1 and g(x) = x².

fg(x) — apply g first

fg(x) = f(g(x))= f(x²)= 2x² + 1

Square first, then ×2 + 1.

gf(x) — apply f first

gf(x) = g(f(x))= g(2x + 1)= (2x + 1)²

×2 + 1 first, then square.

Quick Check — Substitute x = 1

fg(1): 2(1)² + 1 = 3

gf(1): (2(1) + 1)² = 3² = 9

Same x, same functions — different answers. The order changes everything.

Weak vs Strong — Composite Order

The single biggest error on composite questions is reading fg left-to-right. Here's the contrast.

❌ Weak Student

f(x) = 3x, g(x) = x²

Find fg(x):

"Apply f first (left to right)"

f(x) = 3xg(3x) = (3x)² = 9x²

Answer: 9x² ✗

Read fg left-to-right. Computed gf, not fg.

✅ Strong Student

f(x) = 3x, g(x) = x²

Find fg(x):

"Apply g first — the inner one"

g(x) = x²f(x²) = 3(x²) = 3x²

Answer: 3x² ✓

Applied g first, then f. Correct order.

The Mnemonic

"The inner one goes first."

In fg(x), the inner function is g — closest to the x. Apply g first, then f. Always.

Worked Example 5 — Combined Inverse (Past Paper)

Based on AQA Jun 22 Paper 2, Q25. A 4-mark question combining inverse-finding with algebraic simplification.

05Show simplifies to integer [4 marks]

f(x) = 2x + 5. Show that 3f(x) − 12f⁻¹(x) simplifies to an integer.

M — Method. Compute 3f(x) and 12f⁻¹(x) separately. Subtract. If all variable terms cancel, what remains is the integer.

C — Calculate.

3f(x) = 3(2x + 5) = 6x + 15

← M1: compute 3f(x)

Find f⁻¹: y = 2x + 5

← step 1 of inverse procedure

Swap: x = 2y + 5 → y = (x − 5)/2

← M1: correct inverse

12f⁻¹(x) = 12 × (x − 5)/2 = 6(x − 5) = 6x − 30

← simplify

3f(x) − 12f⁻¹(x) = (6x + 15) − (6x − 30)

← M1: subtract

= 45

← A1: x terms cancel

E — Evaluate.3f(x) − 12f⁻¹(x) = 45, which is an integer, as required. ✓

Common Errors to Avoid

Students Often WriteCorrect VersionWhy It Matters

fg(x) = f(x) × g(x)fg(x) = f(g(x)) — composition, not multiplicationCompletely different operation

fg(x) applies f first (left-to-right)fg(x) applies g first (inner-first)Gets gf(x) instead of fg(x)

Inverse of 3x + 2 is 2x + 3 (symbol-flipping)Inverse is (x − 2)/3 — found by proper algebraSymbol-flipping gets wrong answer

f⁻¹(x) read as 1/f(x)f⁻¹(x) is the inverse function, NOT the reciprocalThe −1 is an "undo" marker, not an exponent

Don't check f(f⁻¹(x))Always check: f(f⁻¹(x)) should equal xCatches most algebraic errors

Skip brackets when substituting g(x)Wrap g(x) in brackets: f(g(x)) = f((2x+4))Prevents expansion errors

Examiner Alerts — The Big Ones

Composite Order (Left-to-Right Misread)

What students write: On fg(x) = f(3x) when actually g(x) = x² (they apply f first).

Why it loses marks: this computes gf(x), not fg(x). The two functions are generally different.

What to write instead: "fg(x) means f of g(x). The inner function — g — applies first."

Inverse by Symbol-Flipping

What students write: inverse of 3x + 2 written as 2x + 3 — they flipped the digits and operations randomly.

Why it loses marks: that's not an inverse — if you apply both, you won't get x.

What to write instead: use the 3-step procedure. Write y = …, swap x ↔ y, solve for y. Verify by computing f(f⁻¹(x)).

f⁻¹(x) Read as Reciprocal

What students write: f⁻¹(x) = 1/f(x) = 1/(3x + 2).

Why it loses marks: the superscript −1 here is function notation for the inverse — it is not an exponent. 1/f(x) is the reciprocal, a completely different thing.

What to write instead: treat f⁻¹ as a notation for "the inverse function of f", and find it using the swap-and-solve procedure.

Exam Technique

Top Exam Tip

Before you compute anything, decide which function applies first. For fg(x), that's g. For f⁻¹f(x), it's f (the inside function). Writing the inner function's formula as your first line prevents 90% of composite errors.

Composite Checklist

Did I identify the inner function? In fg(x), the inner is g.

Did I wrap the inner function in brackets? f((g(x))) avoids expansion slips.

Did I expand carefully? Watch for signs when substituting a polynomial.

Did I simplify fully? Collect like terms at the end.

Inverse Checklist

Did I start with y = f(x)? Always write this first line.

Did I swap every x and y? Both directions.

Did I rearrange using standard techniques? (See Lesson 2.07)

Did I check by computing f(f⁻¹(x))? Must equal x.

Practice Question

Combined inverse/composite question

f(x) = 4x − 3 and g(x) = x² + 1.

(a) Find f⁻¹(x). [2 marks]

(b) Work out f⁻¹(17). [1 mark]

(c) Find gf(x) and simplify fully. [3 marks]

(a)

y = 4x − 3. Swap: x = 4y − 3. Add 3: x + 3 = 4y. Divide: y = (x + 3)/4.

(a)

f⁻¹(x) = (x + 3)/4

(b)

f⁻¹(17) = (17 + 3)/4 = 20/4 = 5

(c)

gf(x) = g(f(x)) = g(4x − 3) — apply f first (inner)

(c)

= (4x − 3)² + 1

(c)

= (16x² − 24x + 9) + 1

(c)

gf(x) = 16x² − 24x + 10

Quick-Fire Checklist

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

Can You...

Find f⁻¹(x) when f(x) = 5x − 2? (Answer: (x + 2)/5)

Verify an inverse by computing f(f⁻¹(x))?

State which function applies first in fg(x)? (Answer: g — the inner one)

Find fg(x) when f(x) = 2x and g(x) = x + 3? (Answer: 2(x + 3) = 2x + 6)

Explain why fg(x) ≠ gf(x) in general?

Distinguish f⁻¹(x) from 1/f(x)? (inverse function vs reciprocal — not the same)

Expand (2x + 4)² correctly? (Answer: 4x² + 16x + 16)

One-Line Takeaway

Inverse: swap x ↔ y, solve for y. Composite: the inner function goes first. That's the whole lesson. Everything else is careful algebra from Lessons 2.02, 2.06, and 2.07.

Summary

Key Points

Inverse function f⁻¹(x) — undoes what f does. If f(a) = b, then f⁻¹(b) = a.

Finding f⁻¹ — 3-step procedure: write y = f(x), swap x ↔ y, solve for y.

Key identity — f(f⁻¹(x)) = x. Use this to check your inverse.

Symbol-flipping doesn't work — "inverse of 3x + 2 is 2x + 3" is wrong. Do the algebra.

Composite function fg(x) — means f(g(x)). Apply g first, then f.

Read right to left — in fg(x), the inner function (g) applies first.

fg(x) ≠ gf(x) — order matters. The two are generally different functions.

f⁻¹(x) is not 1/f(x) — inverse function vs reciprocal. Different concepts.

Exam Tip

Combined inverse/composite questions (like Jun 22 P2 Q25 and Nov 23 P1 Q21) are worth 4–5 marks and look intimidating — but they're always two separate skills glued together. Handle each part with its own procedure, then combine the results. Break it into pieces and the mark scheme gives you credit for each step.