A function is a rule with a name
f(x) = 3x − 5 is a rule called f. Whatever goes in the bracket replaces every x in the rule.
- f(4) = 3 × 4 − 5 = 7
- f(−2) = 3 × (−2) − 5 = −6 − 5 = −11
Brackets round a negative input stop sign slips, especially with powers: if g(x) = x² + 1, then g(−3) = (−3)² + 1 = 10, not −8.
A function can also be given in words: "f multiplies a number by 4, then subtracts 7" is f(x) = 4x − 7. "Subtract 7, then multiply by 4" is f(x) = 4(x − 7): the order matters.
An expression in the bracket
The input can be an expression. Put it in brackets, then simplify.
- f(x) = 3x − 5
- f(a + 2) = 3(a + 2) − 5
- = 3a + 6 − 5 = 3a + 1
With f(x) = x² + 4x, f(x − 1) = (x − 1)² + 4(x − 1) = x² − 2x + 1 + 4x − 4 = x² + 2x − 3.
The mistakes to avoid
- f(a + 2) is not f(a) + 2. Replace x by the whole bracket.
- f(x) = 10 is an equation: you are given the output and need the input. Solve 3x − 5 = 10 to get x = 5. Working out f(10) answers a different question.
- f(2x) is not 2f(x): only the x changes.