GCSE · Maths · AQA · Spec 8300 · Higher

Inverse and composite functions (Higher)

Put 3 through 'add 5', then 'double': you get 16. Swap the machines and you get 11. Function notation tells you which order you mean, and how to go backwards.

Follow one number through the machines

Stage 1 of 6: Input 3. paused

Input 3 · 1/6drag to scrub →

Start with x = 3 and work out fg(3). Written as f(g(3)), g sits on the inside, next to the 3, so g gets the number first.

g(x) = x + 5 and f(x) = 2x. Scrub to push 3 through fg, then bring it home.

Read the notation

What is the notation telling you to do?

Same two machines as above: f(x) = 2x and g(x) = x + 5. This time the input is 4.

Only one of these is right. Which would you have said?

fg(4) = 13. f is written first, so f doubles the 4 to make 8, then g adds 5.fg(4) = 72. fg means f(4) times g(4), so 8 × 9.fg(4) = 18. g gets the 4 first and gives 9, then f doubles the 9.f⁻¹(4) = 1/8. The ⁻¹ means one over f(4).

How sure are you?

I'm sureI think soI'm guessing

fg(4) = 13. f is written first, so f doubles the 4 to make 8, then g adds 5.

This is a common misconception

This reads fg like a sentence, left to right. But f can't act on the 4. Its input is g(4), and that doesn't exist until g has run. Doing f first actually works out gf(4) = 13, which is a different function. For fg(4): g(4) = 9, then f(9) = 18.

fg(4) = 72. fg means f(4) times g(4), so 8 × 9.

This is a common misconception

In function notation, fg doesn't mean multiply. It means one machine feeds the next. There is no f(4) here at all: the 4 goes into g, and g's output, 9, goes into f. So fg(4) = f(9) = 18.

fg(4) = 18. g gets the 4 first and gives 9, then f doubles the 9.

This matches the evidence

Yes. fg(4) means f(g(4)). g is the inner function, so it acts first: g(4) = 9. That 9 is f's input, so f(9) = 18. Every function needs an input before it can act, and here f's input is whatever g produces.

f⁻¹(4) = 1/8. The ⁻¹ means one over f(4).

This is a common misconception

With plain numbers, 8⁻¹ does mean 1/8, so this is a natural guess. But on a function, ⁻¹ means the inverse: the machine that undoes f. f doubles, so f⁻¹ halves, and f⁻¹(4) = 2. Check: f(2) = 4, so f⁻¹ takes the output 4 back to the input 2.

fg(x) = f(g(x)): the function next to the x acts first. And f⁻¹ is a reverse machine, not a reciprocal.

Write the chain as one expression

Problem

f(x) = 2x and g(x) = x + 5. Write fg(x) as a single expression, then check it against the machine chain.

Maths · Algebra

Work backwards from an output

You know what came out of the composite. Solve to find what went in.

Goalf(x) = 2x and g(x) = x + 5. Solve fg(x) = 30.

fg(x) = 2x + 10

Start from the composite written as one expression.

Step 1 of 5

Start from the composite written as one expression.

Watch out: Solve the whole composite, not just the outer function. Solving 2x = 30 (f on its own) gives 15, which is g's output, not the input you were asked for.

Find an inverse by changing the subject

y = 3x + 2

Call f's output y. Read the machines: f(x) = 3x + 2 multiplies by 3, then adds 2.

1 / 5

Spot the slip

Where does this inverse go wrong?

f(x) = 5x − 3. Find f⁻¹(x), then use it to find the input that gives an output of 12.

A student's answer — which line goes wrong?

f multiplies by 5, then subtracts 3.
To reverse it, undo the × 5 first: divide by 5.
Then undo the − 3: add 3.
So f⁻¹(x) = x/5 + 3
f⁻¹(12) = 12/5 + 3 = 5.4

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

Chain two function machines together, then run the chain backwards.

What you need to know

f(x) is the output when x goes into f. If f(x) = 2x, then f(7) = 14.
fg(x) means f(g(x)): g acts first, and its output is f's input.
To write fg(x) as one expression, substitute the whole of g(x) for x in f(x).
If you know a composite's output, form an equation and solve it to find the input.
f⁻¹(x) is the inverse of f: it maps each output back to its input. Find it by making x the subject of y = f(x).

The big picture

A function turns an input into an output. A composite function applies two functions in succession: in fg(x), g acts first and its output becomes f's input, so it can't be worked from left to right. You can write a composite as one expression, and solve it to find the input that gives a known output. The inverse function f⁻¹(x) reverses f, taking each output back to its input, and you find it by changing the subject.

Key points

1A function is a rule: one input in, one output out.

2A composite is two functions in succession. Work from the inside out: in fg(x), g first, then f.

3fg(x) and gf(x) are usually different functions.

4To solve fg(x) = k, write fg(x) as one expression, set it equal to k and solve.

5An inverse undoes the operations in reverse order. f⁻¹(x) is not 1 ÷ f(x).

6To check an inverse, put a number through f and then f⁻¹: you should get your number back.

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Worked example

Problem

f(x) = 3x + 1 and g(x) = x². (a) Find gf(x). (b) Solve gf(x) = 49.

⚠ Watch out

Working fg(x) from left to right. f is written first, but it can't act until it has an input, and its input is g(x). So g goes first. Doing f first gives gf(x), which is usually a different function.

🧠

Memory hook

Socks on, then shoes. Shoes off before socks. A composite puts the inside function on first (in fg(x), that's g), and an inverse takes things off in reverse: undo the last step first.

✓

Check yourself

f(x) = x − 7 and g(x) = 3x. Find fg(5), then find f⁻¹(x). (Answers: g(5) = 15, so fg(5) = f(15) = 8. f subtracts 7, so f⁻¹(x) = x + 7.)

Flashcards

(12)

What is a function?

A rule that turns each input into an output, like a machine: a number goes in, a number comes out.

What does f(5) mean?

The output you get when 5 is put into the function f.

What is a composite function?

Two functions applied one after the other: the output of the first becomes the input of the second.

In fg(x), which function acts first, and why?

g. fg(x) means f(g(x)), so f's input is g's output, and that has to exist before f can act.

Are fg(x) and gf(x) the same?

Usually not. Swapping the order changes which function acts first, and usually changes the output.

How do you write fg(x) as a single expression?

Replace every x in f(x) with the whole expression for g(x), in brackets, then simplify.

You're told fg(x) = k. How do you find x?

Write fg(x) as one expression, set it equal to k, and solve the equation.

What is an inverse function?

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

Does f⁻¹(x) mean 1 ÷ f(x)?

No. On a function, ⁻¹ means the inverse (the reverse process), not a reciprocal.

How do you find f⁻¹(x) algebraically?

Write y = f(x), rearrange to make x the subject, then rewrite it with x as the input: f⁻¹(x) = …

When undoing a function, which operation do you undo first?

The last one f did. Reverse the order as well as each operation, like taking shoes off before socks.

How can you check an inverse is right?

Put a number into f, then put the output into f⁻¹. You should get your original number back.

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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Follow one number through the machines

What is the notation telling you to do?

Work backwards from an output

Where does this inverse go wrong?

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