Functions questions in Paper 1 combine algebra with careful language. A student can find an inverse perfectly and still lose a mark for the domain, or can describe a range correctly and lose it for the notation. This guide collects what Cambridge examiners say students get wrong, based on recent reports.
Summarised in our own words from a reading of the examiner reports.
Mistakes that cost marks
- Domain and range confused. This is the most repeated problem. The domain is the set of inputs, the range is the set of outputs. When asked for the range, students write x instead of y or f(x), or use the wrong inequality sign, or give only one end of the range.
- The sign of the root in an inverse. To find an inverse that involves a square root, students must decide between the positive and the negative root. Reports say that this choice is tied to the domain of the original function: the range of the inverse is the domain of the original. Many students lost a mark by choosing the wrong sign, or by leaving ± in the answer.
- Stopping at "y =". After rearranging to make x the subject, the inverse must be written as f⁻¹(x) in terms of x. A few students leave the answer as y =.
- Square roots that do not cover the whole expression. Examiners have to be convinced that a square root sign covers a whole fraction or a whole expression. Write brackets, or draw the root sign clearly.
- Composite functions. The order is mixed up (fg instead of gf), or one function is multiplied by the other. Reports mention students who found the composite incorrectly because they did not multiply the constant inside the function by the outer factor.
- One-to-one and inverse questions. Many students did not realise that a function with a minimum point cannot be one-to-one, so it cannot have an inverse over that domain. Others gave the incorrect statement that a composite function was not one-to-one. When asked to explain why a function is increasing, the derivative was often stated to be positive without justification.
- Mirror line missing in sketches. Sketches of a function and its inverse should include the line y = x and show the reflection clearly.
- Algebra under pressure. Errors in expanding brackets, losing a term, or forgetting to multiply the "1" when clearing a denominator are common when equations are formed from functions.
What examiners advise
- Write the domain or range as a precise statement. A good habit is to say "range of f is f(x) ≥ …" with the correct inequality.
- Link the range of f to the domain of g. For a composite function gf to exist, the range of f must lie inside the domain of g. Reports show that this explanation is often given with the two words swapped.
- Use the answer from an earlier part when the question says "hence" or asks for fg or gf again.
- Give the answer in the form asked. When the question asks for the inverse, write f⁻¹(x) =, not y =.
Habits that help
- Check an inverse by composing it with the original: f(f⁻¹(x)) should give x.
- Use a quick sketch to decide the domain and the sign of the root.
Before you move on
- Is each domain and range written with the right variable and the right inequality?
- Did I choose the sign of the root using the domain?
- Is the inverse written as a function of x?
- Did I use the correct order for composite functions?
Questions students ask
What is the difference between domain and range, and why do students mix them up?
The domain is the set of inputs and the range is the set of outputs. Examiners say this is the most repeated problem in functions. When the range is asked for, students write x instead of y or f(x), use the wrong inequality sign, or give only one end of the range.
How do I write the range of a function properly?
Write a precise statement such as "range of f is f(x) ≥ …" with the correct inequality, using y or f(x), not x.
How do I choose the positive or negative square root when finding an inverse?
Use the domain of the original function. The range of the inverse is the domain of the original, so that tells you which sign to choose. Students lose marks by choosing the wrong sign or by leaving ± in the answer.
Should I leave my inverse function as y = something?
No. After making x the subject, write the answer as f⁻¹(x) in terms of x. A few students leave it as y =, and examiners ask for the function form when the question asks for the inverse.
How do I know whether a function has an inverse?
It must be one-to-one. Examiners report that many students did not realise a function with a minimum point cannot be one-to-one, so it has no inverse over that domain. A quick sketch helps you decide.
What do I need for a composite function such as gf to exist?
The range of f must lie inside the domain of g. Reports show that students often give this explanation with the two words swapped, and that they mix up the order of fg and gf.
Practice this topic
Ready to try some? Work through past paper questions on this topic one at a time. Attempt each question fully first, and open the mark scheme only afterwards.