This module covers functions and mappings: a rule that takes an input and gives exactly one output. You will evaluate functions, combine them, reverse them, find which inputs are allowed and read them inside real situations. The same ideas return in graphs, transformations and algebraic fractions.
Check the current Cambridge IGCSE Mathematics 0580 syllabus for the exact content points and notation in your exam year. Our Mathematics learning guide shows where this module sits among the others.
What should you already know?
You need to substitute numbers into algebraic expressions, work with negative numbers and expand a simple bracket. Solving a one-step or two-step linear equation is also needed, because inverse functions rely on it. If any of these feel shaky, revise them first.
An orienting example
Let f(x) = 2x + 3 and g(x) = x². Find f(−4), gf(−4) and f⁻¹(−5).
Step 1, evaluate: f(−4) = 2(−4) + 3 = −8 + 3 = −5.
Step 2, compose: gf(−4) means f first, then g. We have f(−4) = −5, so g(−5) = (−5)² = 25.
Step 3, invert: y = 2x + 3, so x = (y − 3)/2, and f⁻¹(x) = (x − 3)/2. Then f⁻¹(−5) = (−5 − 3)/2 = −8/2 = −4.
Check: f(−4) = −5 and f⁻¹(−5) = −4. The inverse returns the original input, which confirms it.
That one question used four of the five lessons: evaluation with a negative input, composition, inversion and careful signs. It shows how the module hangs together.
In which order should you study it?
- Evaluate a function with negative inputs: the substitution habit that protects every sign.
- Trace an input through a composite function: which function goes first, and why the order matters.
- Reverse a one-to-one mapping: undoing a rule step by step and checking with a number.
- Restrict inputs to keep an expression defined: the two conditions that exclude values from a domain.
- Interpret a function machine in context: a delivery charge that uses all the earlier skills at once.
Then work through the mixed practice set. One lesson a day and the practice set at the weekend is a steady pace.
Which traps catch most students here?
- Losing the bracket and calculating −3² instead of (−3)².
- Reading fg as “f first” when it means g first.
- Taking the reciprocal and calling it the inverse.
- Undoing operations in the wrong order when reversing.
- Listing one restriction when a fraction and a root are both present.
- Ignoring the domain in a context question.
Each lesson shows one of these slips in full and then corrects it.
How should you use the practice set?
Attempt each question on paper before opening the answer. Write the working as you would in an exam. The function composition and inverse explorer lets you test a rule or a composite with any input, and the mistake log and retest queue helps you track the slips that repeat.
When you get something wrong, read the routing table at the end of the practice set and return to the lesson it names. Fix the lesson, then retry a similar question a few days later.