Understanding Functions and f(x) Notation Revision Notes | GCSE AQA Higher Further Maths

1. What Is a Function?

Every time you type a number into a calculator and press a button, something predictable happens: one input produces one output. That predictability is exactly what a function is.

Think of a vending machine: you press B3 and you always get the same snack. Press B3 twice and you get the same result both times. That one-input-one-output guarantee is the defining feature of a function.

The machine above takes any number $x$, squares it, then subtracts 9. Feed in 3, get 0. Feed in 5, get 16. The rule never changes — that consistency is what makes it a function.

2. f(x) Notation — Reading and Writing It

Mathematicians use a compact shorthand to describe function rules. Instead of writing 'the rule that squares a number and subtracts 9', they write $f(x) = x^2 - 9$. The letter $f$ names the function; the $(x)$ shows what the input variable is called.

When you see $f(3)$, it means 'evaluate the function when $x = 3$'. You replace every $x$ in the rule with 3 and calculate the result.

3. Evaluating a Function — Step-by-Step

Now that you can read $f(x)$ notation, the key skill is evaluating — substituting a specific input and working out the output. Here is the method applied to the specification example $f(x) = x^2 - 9$.

Worked example: Evaluating f(x) = x^2 - 9 at x = 5

Given $f(x) = x^2 - 9$, find $f(5)$.

1. $f(x) = x^2 - 9$ — Start by stating the rule clearly before substituting. [M1]
2. $f(5) = 5^2 - 9$ — Replace every $x$ with 5. The brackets tell you the input is 5. [M1]
3. $5^2 = 5 \times 5 = 25$ — Always expand the power longhand to avoid errors.
4. $f(5) = 25 - 9 = 16$ — Subtract 9 from 25. [A1]

Answer: $f(5) = 16$

Notice that the output is a single number, 16. The notation $f(5) = 16$ is a complete, precise statement: when the input is 5, the output is 16.

Worked example: Evaluating f(x) = x^2 - 9 at a Negative Input

Given $f(x) = x^2 - 9$, find $f(-3)$.

1. $f(x) = x^2 - 9$ — State the rule before substituting.
2. $f(-3) = (-3)^2 - 9$ — Use brackets around $-3$ when substituting — this is critical for negative inputs. [M1]
3. $(-3)^2 = (-3) \times (-3) = 9$ — A negative number squared gives a positive result. Expanding longhand prevents sign errors.
4. $f(-3) = 9 - 9 = 0$ — The output is 0. [A1]

Answer: $f(-3) = 0$

The result $f(-3) = 0$ tells you something interesting: $x = -3$ is an input that produces an output of zero. You will use this idea when solving equations involving functions.

4. Visualising f(x) = x^2 - 9

Every function rule has a graph. For $f(x) = x^2 - 9$, the graph is a U-shaped parabola. The points where the curve crosses the $x$-axis are the inputs that give an output of zero — and you have already found one of them: $x = -3$.

The curve crosses the $x$-axis at $x = -3$ and $x = 3$ — both inputs give $f(x) = 0$. The lowest point of the curve is at $(0, -9)$, which is where $f(0) = 0^2 - 9 = -9$.

5. Solving Equations Using f(x) Notation

Evaluating a function means finding the output for a known input. The reverse task — finding the input when you know the output — means solving an equation. Both skills use the same notation, so it is worth being clear about which direction you are working in.

Worked example: Solving f(x) = 7 for f(x) = x^2 - 9

Given $f(x) = x^2 - 9$, solve $f(x) = 7$.

1. $x^2 - 9 = 7$ — Replace $f(x)$ with the rule and set it equal to 7. [M1]
2. $x^2 = 7 + 9 = 16$ — Isolate $x^2$ by adding 9 to both sides.
3. $x = \pm\sqrt{16} = \pm 4$ — Remember both the positive and negative square root. [A1 for both solutions]
4. $x = 4$ or $x = -4$ — Both inputs give an output of 7. Verify: $f(4) = 16 - 9 = 7$ and $f(-4) = 16 - 9 = 7$. [A1]

Answer: $x = 4$ or $x = -4$

Earlier, the graph showed the curve is symmetric about the $y$-axis. That symmetry explains why solving $f(x) = 7$ gives two solutions — one positive and one negative.

6. Practice — Consolidating the Method

You have seen the method applied to $f(x) = x^2 - 9$. Now try these problems yourself before checking the solutions. Work through each substitution step by step rather than computing everything in one go.

7. Exam Technique and Common Mistakes

Function notation questions are reliable mark-earners once you know the traps. The errors below appear repeatedly in mark schemes — recognising them now means you will not lose marks in the exam.

Mastering $f(x)$ notation means you can define any function rule concisely, evaluate it for any input, and solve equations that ask for the input given the output — all three skills are directly assessed at GCSE Further Maths level.