GCSE · Maths · AQA · Spec 8300 · Foundation

Functions — inputs and outputs

Put 7 into a machine and 3 comes out. Could you get back to the 7 from the 3 alone? And what if another machine is plugged into the front?

Put 7 into f(x) = (2x + 1)/5

Stage 1 of 5: In: 7. paused

In: 7 · 1/5scrub 7 through →

The number you feed into the machine is the input. Here x = 7.

Every input goes through the same operations, in the same order, and comes out as exactly one output.

UK note

AQA Higher tier: you are expected to understand and use f(x) notation, and the fg(x) and f⁻¹(x) notation further down this page. At Foundation tier, the part you need is reading an expression as a machine: input, operations in order, output.

Higher

Inverse function

Run the machine backwards

f sends x through × 2, then + 1, then ÷ 5, so f(7) = 3. The inverse function f⁻¹ takes an output of f back to the input that made it. Put the steps of f⁻¹ in the order it applies them.

1 · First step of f⁻¹3 · Last step of f⁻¹

− 1
undoes the + 1
÷ 2
undoes the × 2
× 5
undoes the ÷ 5

Higher

Composite function

Two machines joined: which one runs first?

f(x) = (2x + 1)/5 is the machine from the top of the page, and g(x) = x + 5. The composite function fg(x) joins the two machines into one.

How do you work out fg(2)?

Read it left to right: work out f(2) first, then put that answer into g.Work out g(2) first, then put that answer into f.Work out f(2) and g(2) separately, then multiply them.The order doesn't matter: g then f gives the same answer as f then g.

How sure are you?

I'm sureI think soI'm guessing

Read it left to right: work out f(2) first, then put that answer into g.

This is a common misconception

Reading left to right feels natural, but it runs the machines the wrong way round. f(2) = 5 ÷ 5 = 1, then g(1) = 6. That is gf(2), a different composite. In fg(2) the g sits next to the 2, so the 2 goes into g first.

Work out g(2) first, then put that answer into f.

This matches the evidence

Yes. fg(2) means f(g(2)): the function next to the input acts first. g(2) = 7, then f(7) = 3, which is the run you scrubbed through the machine at the top of the page. So fg(2) = 3.

Work out f(2) and g(2) separately, then multiply them.

This is a common misconception

Writing f and g side by side looks like multiplying, the way 2x means 2 × x. But f(2) × g(2) = 1 × 7 = 7, and that is a product, not a composite. In a composite, one machine's output is fed into the next machine. Nothing is multiplied.

The order doesn't matter: g then f gives the same answer as f then g.

This is a common misconception

Try both. g then f gives f(7) = 3. f then g gives g(1) = 6. The answers are different, so the order is part of what fg means.

In fg(x), g acts first and its output becomes the input to f: fg(2) = f(g(2)) = f(7) = 3.

Higher

From machine to algebra

Write f⁻¹(x) and fg(x) as expressions

f(x) = 3x − 4 and g(x) = 2x + 5. (a) Find f⁻¹(x). (b) Find fg(x) as a single expression.

(a) Let y = 3x − 4, so y is the output when the input is x.
Undo the − 4 by adding 4 to both sides: y + 4 = 3x
missing step

Which line is step 3?

x = (y + 4)/3x = y/3 + 4x = y + 4/3

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

One machine, three ways: run it forwards, run it backwards, or plug it into another machine.

What you need to know

A function takes each input through the same operations, in a fixed order, and gives exactly one output.
Read the order from the expression: in (2x + 1)/5, multiply by 2, then add 1, then divide the whole numerator by 5.
Higher: f(x) names the output for input x, so f(7) = 3 means that input 7 gives output 3. It does not mean f × 7.
Higher: the inverse function f⁻¹ runs the machine backwards. Undo each operation with its opposite, in reverse order, and each output of f goes back to the input that made it.
Higher: fg(x) = f(g(x)) is a composite function. g, next to x, acts first and its output is the input to f. It is not f(x) × g(x), and gf(x) is usually a different function.
Higher: to write f⁻¹(x), set y = f(x) and rearrange for x. To write fg(x), put the whole of g(x), in a bracket, in place of x in f.

The big picture

A function takes an input, applies its operations in a fixed order and gives one output, and f(x) names that output. Running the same machine backwards gives the inverse function f⁻¹(x), and joining two machines gives a composite function such as fg(x), where g acts first. Order decides all three.

Key points

1Input → operations in a fixed order → one output.

2Higher: f(7) = 3 means 'put 7 in, get 3 out', not f × 7.

3Higher: f⁻¹ uses the opposite operations in reverse order.

4Higher: in fg(x), g acts first; fg(x) = f(g(x)).

5Higher: f⁻¹(x) is not 1/f(x).

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Worked example

Problem

f(x) = 4x + 1 and g(x) = x². (a) Find f(−2). (b) Find gf(x). (c) Find f⁻¹(x).

⚠ Watch out

Finding an inverse by swapping each operation for its opposite but keeping the forward order. For f(x) = (2x + 1)/5 that gives ÷ 2, then − 1, then × 5, which sends 3 to 2.5 instead of back to 7. Undo the last operation first.

🧠

Memory hook

One machine, three ways. Forwards is the function. Backwards is the inverse: shoes off before socks, so the last thing done is the first thing undone. Joined is the composite: the machine next to x goes first.

✓

Check yourself

Without looking back: h(x) = 2x − 9. Work out h(6). Then write h⁻¹(x), and check that it sends your answer back to 6.

Flashcards

(9)

What does a function do to each input?

It applies the same operations, in a fixed order, and gives exactly one output.

(Higher) What does f(x) mean?

The output of the function f when the input is x. f(7) = 3 means input 7 gives output 3. It does not mean f × 7.

For the expression 3(x − 2), in what order do the operations act on x?

Subtract 2 first, because it is inside the bracket, then multiply by 3.

(Higher) What is the inverse function f⁻¹?

The function that reverses f: it takes each output of f back to the input that produced it.

(Higher) How do you build the inverse machine from the machine for f?

Use the opposite of each operation, in reverse order, so f's last operation is undone first.

(Higher) What is a composite function?

Two functions applied in succession: the output of the first becomes the input of the second.

(Higher) In fg(x), which function acts first?

g, the one written next to x. fg(x) = f(g(x)).

(Higher) Does f⁻¹(x) mean 1/f(x)?

No. The ⁻¹ means inverse: f⁻¹ undoes f. 1/f(x) is the reciprocal of f(x), which is a different thing.

(Higher) How can you check an inverse you have found?

Put a number through f, then put the output through f⁻¹. You should get the number you started with.

Tap any card to flip it, or use Study as deck to go through them one at a time. In the full lesson these run as a spaced-repetition deck — you rate each card Hard, Good or Easy and the tricky ones keep coming back until they stick.

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Put 7 into f(x) = (2x + 1)/5

Run the machine backwards

Two machines joined: which one runs first?

Write f⁻¹(x) and fg(x) as expressions

More AQA GCSE Maths topics