Functions

Inputs and outputs

• A function is a rule that takes a number in and gives exactly one number back
• The number going in is the input and the number coming back is the output
• Writing f(x) = 4x − 7 names the rule f and says what it does to whatever is put in
• Any letter can name a function, and a question that needs a second rule simply uses a second letter such as g or h
• Every x in the rule is replaced by whatever sits inside the brackets, whether that is a number, a letter or a whole expression
  • f(3) = 4 × 3 − 7 = 5
  • f(−2) = 4 × (−2) − 7 = −15
  • f(a) = 4a − 7, which cannot be tidied any further
• Bracket a negative input as you substitute it, otherwise the sign is easily dropped

Reading the question the right way round

• f(6) means put 6 in and see what comes out
• f(x) = 6 means the output is 6 and the input is unknown, so it is an equation to solve
• These two look almost identical on the page and mean opposite things, so read which side the number is on
• To solve f(x) = 21 for f(x) = 4x − 7, replace f(x) by the rule: 4x − 7 = 21, so 4x = 28 and x = 7
• A mapping diagram shows the same idea as two columns joined by arrows, inputs on the left and outputs on the right, with one arrow leaving each input
  • For f(x) = x + 3 the arrows run 3 → 6, 4 → 7 and, in general, x → x + 3