Function Notation
Function notation is the language the rest of the functions topic is written in: f(x) is not f times x, it is the output of the rule f when the input is x. This page covers what a function is, how to work out f(k) including when k is negative, the difference between f(4) and f(x) = 4, substituting a whole expression such as f(2x), and reading values off the graph of y = f(x) forwards and backwards. Work through the worked examples, then test every skill on the auto-marked practice questions below.
What Function Notation Means
A function is a rule that turns each input into exactly one output. The rule is given a name, and \(f(x)\) is read as "\(f\) of \(x\)": the output of the rule \(f\) when the input is \(x\).
\(f(x)\) is not \(f\) multiplied by \(x\). The letter names the rule; the bracket holds whatever you are putting in. So if \(f(x) = 2x + 5\), then \(f(3) = 2 \times 3 + 5 = 11\), and the 3 simply goes wherever the \(x\) was.
The same function can be written \(f : x \mapsto 2x + 5\), read "\(f\) maps \(x\) to \(2x + 5\)". It means exactly the same thing; the papers use the \(f(x) = \ldots\) form.
Reading \(f(x)\) off a Graph
On the graph of \(y = f(x)\), the value of \(f(a)\) is simply the height of the curve above \(x = a\).
- 1Going forwards — to find \(f(a)\), go up from \(a\) on the \(x\)-axis until you meet the curve, then across to the \(y\)-axis and read the value.
- 2Going backwards — to solve \(f(x) = c\), go across from \(c\) on the \(y\)-axis, and read off the \(x\)-coordinate of every point where the line meets the curve.
- 3Count the crossings. Going backwards can give two answers, one, or none at all — a horizontal line can cut a curve more than once.
Core Ideas
Worked Examples
💡 Example 1: Working out \(f(k)\) (Room 1)
\(f(x) = 3x - 7\). Find \(f(5)\) and \(f(-2)\).
💡 Example 2: Finding the input (Room 2)
\(f(x) = 4x + 9\). Solve \(f(x) = 33\).
Nothing was substituted here. \(f(33)\) would have been a different question with a different answer.
💡 Example 3: Substituting an expression (Room 1)
\(f(x) = 5x + 2\). Find \(f(3x - 1)\) in its simplest form.
💡 Example 4: Reading a graph both ways (Room 3)
The graph of \(y = f(x)\) is shown. Write down \(f(1)\), then solve \(f(x) = 5\).
Two crossings, so two answers. A question that asks you to solve from a graph is telling you to expect more than one.
🔑 Key Points
- \(f(x)\) is the output when the input is \(x\). It is not \(f\) times \(x\).
- \(f(k)\) means substitute; \(f(x) = k\) means solve.
- Keep the bracket when the input is negative: \((-3)^2\) is \(9\), not \(-9\).
- On a graph, \(f(a)\) is the height of the curve above \(x = a\).
- \(f : x \mapsto \ldots\) is the same function written a second way.
⚠️ Common Pitfalls
- Reading \(f(x)\) as a multiplication.
- Substituting the output: working out \(f(4)\) when the question said \(f(x) = 4\).
- Dropping the bracket on a negative input.
- Giving only one answer when the graph crosses the line twice.
- Leaving \(f(2x)\) unexpanded when the question asks for its simplest form.
Ready to practise? Work through all four rooms below.
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