Function questions in Additional Math combine several skills: domain and range, inverse and composite functions, and sketching a function with its inverse. The reports show that students can usually carry out the algebra. The marks are lost in the notation and in what the question actually asks for. 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
- The range and its notation. The most repeated problem. Students used the wrong inequality sign (f(x) > 1.5 instead of f(x) ≥ 1.5), wrote notation such as "x ≥ 1.5", "range ≥ 1.5" or just "1.5", ignored the domain of f when stating the range, or gave an answer unrelated to the function. Common wrong ranges include "the set of real numbers", x > 0 and y ∈ ℝ.
- Domain and range of the inverse. Many candidates did not give the range of the inverse function, and some gave incorrect notation or an incorrect answer. The domain of f⁻¹ is the range of f, and examiners expect the answer as an inequality in terms of x. Reports also describe students who stated the domain of f instead, or whose answer did not follow from their earlier part.
- Finding the inverse. Sign errors, order-of-operations errors, and method errors such as turning e^(−y) = x/5 into −y = −ln(x)/5 (a misuse of the logarithm). Others forgot the brackets around the argument of a logarithm, writing y = log₂ x − 1 instead of y = log₂ (x − 1), or combined several steps and made a sign slip.
- Composite functions. Not realising that g² means g(g), and squaring the expression (as in (x + 1)²). Another common error was the wrong order of operations when solving, such as subtracting 7 from 4 before dealing with the logarithm.
- Sketching a function and its inverse. Omitting the intercepts, sketching only in the first quadrant, wrong shapes, and no points of intersection. Some students did not know the relationship between the two graphs and drew a reflection in the x-axis or the y-axis, when it is a reflection in the line y = x. In one question the final mark required graphs that intersect y = x twice, as the question said.
- Rejecting a value. After finding two solutions, forgetting to reject the one that lies outside the domain.
- "Least value of k." Giving an inequality when the question wanted a single value.
- Contradicting yourself. Writing a correct statement and also an incorrect one, such as "f is one-to-many".
- Answer form. Not writing the answer as a single logarithm when asked, and omitting essential brackets, which caused later errors.
- Misreading the demand. In one question many equated the expression to 0 and solved, when the question needed the stationary (least) value, found by differentiating or completing the square.
What examiners advise
- Read the demand carefully and check that your answer uses all the information in the question.
- Write the range as an inequality in terms of f(x), or y, and the domain of the inverse as an inequality in terms of x, and check the sign of the inequality.
- When you sketch a function and its inverse, show the intercepts, draw them as reflections in y = x, and mark the points where they meet the line.
- Use brackets around the argument of a logarithm and in composite functions.
Habits that help
- Write the range of f first. It gives you both the range of f and the domain of f⁻¹.
- After finding an inverse, check it by substituting a value, such as f(f⁻¹(x)) = x.
- For a composite function, write "g(g(x))" in full before you start.
Questions students ask
How do I write the range of a function in Additional Math?
As an inequality in terms of f(x) or y, such as f(x) ≥ 1.5, and check the sign. Examiners report f(x) > 1.5 used instead of f(x) ≥ 1.5, and notation such as "x ≥ 1.5", "range ≥ 1.5" or just "1.5". It is the most repeated problem.
What is the domain and range of the inverse function?
The domain of f⁻¹ is the range of f, and its range is the domain of f. Many candidates did not give the range of the inverse, or used wrong notation.
What does g² mean?
g(g(x)), not the square of the expression. Write g(g(x)) in full before you start. Students squared the expression as (x + 1)².
What must I show when sketching a function and its inverse?
The intercepts, the two graphs as reflections in y = x, and the points where they meet the line. Students sketched only in the first quadrant, omitted intercepts or drew wrong shapes.
What if the question asks for the "least value of k"?
Give a single value, not an inequality. Also remember to reject a solution that lies outside the domain.
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.