f(x) means "substitute the value for x"
means replace every with 4 and evaluate. Solving means set the expression equal to 10 and solve for .
Composite functions: inside first
(f of g of x — apply g first, then f) means — do first, then feed the result into . The order matters: is usually not the same as .
Inverse: swap and rearrange
To find (the inverse of f), write , swap the roles of and , then make the subject. The inverse undoes the function.
Drawn from real examiner reports.
Cannot factorise to isolate the subject
When the variable appears twice (e.g. ), students did not grasp factorising to collect and isolate the intended subject, so could not finish the inverse.
Highlighted on June 2024 Paper 1H (Q15b).
Keeping the invalid root of an inverse
For an inverse found via completing the square, the answer was given without rejecting the negative root, even though the domain needs only the positive one.
Seen on November 2024 Paper 1H (Q25).
Doing the composite in the wrong order
Computing when was asked (or vice versa) — remember the function nearest the acts first.
Reading f(4) as solve f(x) = 4
These are opposite tasks: means substitute 4 for and evaluate, while solving means set the rule equal to 4 and solve for . Doing one when the other is asked loses the marks.
Only doing one inverse operation
For the inverse must undo both steps — multiply by 2, then subtract 4 — giving . Stopping after one operation (e.g. ) is wrong.
Check an inverse with a number
Pick a value, apply , then apply — you should return to your starting number. It catches rearrangement slips instantly.
Work composites from the inside out
For , evaluate first, then apply to that result. Writing the inner value down before applying the outer function stops order mix-ups.
Respect a stated domain
If a domain such as is given, use it to reject the invalid root of an inverse and keep only the branch that fits — examiners award the final mark for this.
(a named function of x) is a rule. means substitute 4 for . Solving means set the rule equal to 10 and solve. For : ; and .
What does f(4) mean?
The function is given by . (a) Find . (b) Solve .
