Functions as a Mapping

Number machines forwards and backwards, a machine as an expression, function notation f(5), and values that cannot be in the domain.

In the exam for Cambridge IGCSE Maths 0580, Cambridge O Level Maths 4024, Edexcel IGCSE Maths 4MA1 · 58 real past-paper questions below

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Number Machines

A number machine does its steps in order. Follow the arrows.

Worked example

1
$10 \to + 5 \to \times 4 \to ?$ — The machine. It does +5 first, then × 4, in the order the arrows point.
2
$10 + 5 = 15$ — First arrow: add 5 to the input 10.
3
$15 \times 4 = 60$ — Next arrow: multiply the 15 by 4. The output is 60.

Common mistake

Doing 5 × 4 first. The machine goes in order, left to right.

Working Backwards

“Work out the input when the output is 100”: go right to left and undo each step.

Worked example

1
$? \leftarrow - 5 \leftarrow \div 4 \leftarrow 100$ — Draw the arrows backwards from the output 100. Each step becomes its opposite: + 5 becomes − 5, and × 4 becomes ÷ 4.
2
$100 \div 4 = 25$ — Undo the last step first. The last step was × 4, so divide by 4.
3
$25 - 5 = 20$ — Now undo the + 5 by taking away 5. The input is 20.

Key idea

Check forwards: (20 + 5) × 4 = 100.

A Machine as an Expression

When the input is a letter, write each step on the letter. Use a bracket before dividing or multiplying a whole expression.

Worked example

1
$y \to \times 7 \to - 9$ — A machine with the letter y as its input.
2
$7 y$ — First step: y × 7 is written 7y.
3
$7 y - 9$ — Second step: take 9 from that. This is the output.
4
$n \to \times 3 \to + 5 \to \div 2$ — Another machine, with input n.
5
$3 n$ — First step: n × 3 is written 3n.
6
$3 n + 5$ — Second step: add 5 to the 3n.
7
$P = \frac{3 n + 5}{2}$ — Last step: divide by 2. Write it as a fraction, so the whole of 3n + 5 is divided by 2.

Common mistake

Writing 3n + 5 ÷ 2. That only divides the 5.

Function Notation (Higher)

$f (x) = 3 x - 4$ is a machine: times 3, then take 4. $f (5)$ means put 5 in.

Worked example

1
$f (5) = 3 (5) - 4$ — Replace every x with 5, and keep the brackets.
2
$= 15 - 4$ — Multiply first: 3 × 5 = 15.
3
$= 11$ — Then subtract: 15 − 4 = 11.
4
$f (- 2) = 3 (- 2) - 4$ — Replace x with −2. The brackets stop you losing the minus sign.
5
$= - 6 - 4$ — 3 × (−2) = −6, because a positive times a negative is negative.
6
$= - 10$ — −6 − 4 = −10. Both are negative, so add the sizes and keep the minus.
7
$f (\frac{1}{3}) = 3 (\frac{1}{3}) - 4$ — Replace x with 1/3. A fraction works the same way.
8
$= 1 - 4$ — 3 × 1/3 = 1, because the 3s cancel.
9
$= - 3$ — 1 − 4 = −3.

A square needs more care. Square the bracket first, then multiply.

Worked example

1
$f (x) = x^{2} - 3 x$ — A function with a square in it. Find f(−2).
2
$f (- 2) = (- 2)^{2} - 3 (- 2)$ — Replace each x with −2, in brackets.
3
$= 4 - 3 (- 2)$ — (−2)² = 4. A negative times a negative is positive.
4
$= 4 + 6$ — −3 × (−2) = +6, for the same reason.
5
$= 10$ — 4 + 6 = 10.

Common mistake

Writing −2² and getting −4. Without brackets only the 2 is squared. Write (−2)² = 4.

Solving f(x) = a Number (Higher)

Exams often give you the output and ask for the input. Write $f (x)$ equal to the number, then solve it like an equation.

Worked example

1
$f (x) = 3 x - 4, f (x) = 17$ — Find x when the output is 17.
2
$3 x - 4 = 17$ — Write the rule for f(x), and put it equal to the output 17.
3
$3 x = 21$ — Add 4 to both sides.
4
$x = 7$ — Divide both sides by 3.
5
$f (7) = 3 (7) - 4 = 17$ — Check: put 7 back into the machine. It gives 17, so x = 7 is right.

A square in f often gives two answers. Move everything to one side and factorise.

Worked example

1
$f (x) = x^{2} + 2 x, f (x) = 15$ — Find x when the output is 15.
2
$x^{2} + 2 x = 15$ — Write the rule for f(x), equal to 15.
3
$x^{2} + 2 x - 15 = 0$ — Take 15 from both sides, so one side is 0.
4
$(x + 5) (x - 3) = 0$ — Factorise. Two numbers that multiply to −15 and add to +2 are +5 and −3.
5
$x = - 5 or x = 3$ — Each bracket equal to 0 gives an answer.
6
$f (- 5) = 25 - 10 = 15, f (3) = 9 + 6 = 15$ — Check both: (−5)² = 25 and 2 × (−5) = −10; 3² = 9 and 2 × 3 = 6. Both give 15.

Common mistake

Putting 17 in place of x. The 17 is the output, so it goes on the right of the equals sign.

Key idea

Show each line. Most of the marks are for writing f(x) equal to the number and for solving correctly. If the question limits x, for example x > 0, cross out any answer that does not fit.

Numbers That Cannot Go In (Higher)

Some inputs break a function. You cannot divide by 0, and you cannot square root a negative number.

Worked example

1
$g (x) = \frac{1}{3 x + 1}$ — A fraction with x in the bottom. The bottom of a fraction must never be 0.
2
$3 x + 1 = 0$ — Find the input that makes the bottom 0.
3
$3 x = - 1$ — Take 1 from both sides.
4
$x = - \frac{1}{3}$ — Divide both sides by 3.
5
$x \neq = - \frac{1}{3}$ — Only this one value is left out of the domain. Every other input works.

Key idea

For $x - 2$ the inside cannot be negative, so x must be 2 or more.

Which Method? Read the Question

Find what you are given. An input means go forwards. An output means undo backwards. $f (5)$ means put 5 in. $f (x) = 17$ means solve for x.

You see	You do
“Find the output”	Follow the steps in order
“Find the input”	Undo each step, last first
A letter goes in	Write an expression; bracket before ÷
f(5), f(−2)	Replace x, negatives in brackets
“Cannot be in the domain”	Bottom = 0, or root of a negative

Key idea

Reversing a machine is the same idea as an inverse function. See Inverse Functions.

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