What \(f(x)\) means
A function is a rule that turns an input into a single output. Rather than writing "the rule" every time, we give the rule a short name, usually the letter \(f\), and write it as \(f(x)\). You read this aloud as "\(f\) of \(x\)". The letter inside the bracket, \(x\), stands for the input, and the whole thing \(f(x)\) stands for the output.
- Function notation
- A way of writing a rule so its name and its input are both shown. In \(f(x)\), the \(f\) names the rule and the \(x\) is the input you feed in.
So a rule such as "double the number, then add three" is written compactly as
The bracket is the important part. Writing \(f(4)\) does not mean \(f\) multiplied by \(4\). It means "the output of the rule \(f\) when the input is \(4\)". Think of the bracket as a small box that says "put this in".
Where this is assessed
Reading and using \(f(x)\) correctly is Criterion A (knowing and understanding). Laying out your substitution and working clearly is Criterion C (communicating), and using a function to answer a real question reaches into Criterion D (applying maths in context).
Substituting a value
To find an output, you replace every \(x\) in the rule with the input number, then work it out. Nothing more. Using \(f(x) = 2x + 3\) from above, to find \(f(4)\) you swap each \(x\) for \(4\):
Keep the input in brackets while you substitute, especially when it is negative. For \(f(-5)\) you would write \(f(-5) = 2(-5) + 3 = -10 + 3 = -7\). The brackets stop you losing the minus sign.
Read the bracket as "put in"
Whenever you see \(f(\;\text{something}\;)\), read it as "put that something in for \(x\)". This still works when the something is a squared term: for \(g(x) = x^2 - 1\), the output \(g(-3) = (-3)^2 - 1 = 9 - 1 = 8\).
Solving \(f(x)=k\)
Substituting goes forwards: input in, output out. Solving \(f(x) = k\) goes the other way. Here you are told the output \(k\) and asked which input produced it. You set the rule equal to \(k\) and solve the equation for \(x\).
Suppose \(f(x) = 2x + 3\) again and you want the input that gives an output of \(17\). Write \(f(x) = 17\), which means \(2x + 3 = 17\). Solve it: subtract \(3\) to get \(2x = 14\), then divide by \(2\) to get \(x = 7\). You can check by substituting back: \(f(7) = 2(7) + 3 = 17\), which is exactly the output you wanted.
A function is \(f(x) = 4x - 5\). Find \(f(3)\), then solve \(f(x) = 23\).
Check yourself
Try these before opening the answers. Show the working, not just the final value.
1. If \(f(x) = 3x + 2\), find \(f(5)\). +
Replace every \(x\) with \(5\): \(f(5) = 3(5) + 2 = 15 + 2\). So \(f(5) = 17\).
2. If \(g(x) = x^2 - 1\), find \(g(-3)\). +
Keep the input in brackets: \(g(-3) = (-3)^2 - 1 = 9 - 1\). So \(g(-3) = 8\).
3. If \(f(x) = 2x - 7\), solve \(f(x) = 11\). +
Set the rule equal to \(11\): \(2x - 7 = 11\). Add \(7\) to both sides: \(2x = 18\). Divide by \(2\): \(x = 9\). Check: \(f(9) = 2(9) - 7 = 11\). So \(x = 9\).
Part of the Standard Maths library. Spotted an error or want a topic added? That feedback makes the notes better.