The inverse of a one-to-one function swaps inputs and outputs. Its domain equals the original function’s range, while its range equals the original domain.
Quick answer: Check one-to-one behaviour, swap domain and range, select the correct inverse branch, and verify composition on the permitted domains.
Why the distinction matters
The inverse of a one-to-one function swaps inputs and outputs. Its domain equals the original function’s range, while its range equals the original domain.
The mistake is often one of interpretation rather than arithmetic. Students need to know what the question requests, what the mathematical expressions mean and which restrictions apply.
Worked example 1
For f(x)=3x−2 on all real x, f⁻¹(x)=(x+2)/3. Both functions have all-real domains and ranges, and f(f⁻¹(x))=x.
Worked example 2
For f(x)=(x−2)²+1 restricted to x≥2, the range is y≥1. The inverse is f⁻¹(x)=2+√(x−1), with domain x≥1 and range y≥2.
Worked example 3
For the same quadratic restricted instead to x≤2, the inverse is f⁻¹(x)=2−√(x−1). Its domain remains x≥1 but its range is y≤2. The original restriction determines the square-root branch.
The mathematical rule
Check one-to-one behaviour, swap domain and range, select the correct inverse branch, and verify composition on the permitted domains.
How students can repair the mistake
Keep the original attempt, locate the first incorrect decision, and explain why the correct method applies. After a guided example, solve a changed question without hints and check it in the original conditions.
Who should students ask?
The school Mathematics teacher is the first contact for the current topic and assessment expectations. Bring the actual question and the first uncertain line. A tutor can help with persistent gaps after school clarification.
How parents can help
Ask what the student can explain independently rather than how many worksheets have been completed. Support a realistic study timetable and encourage an honest discussion of mistakes.
A practical one-week plan
Identify the error, review its prerequisite, solve a changed example and revisit the method after several days. Do not add unnecessary timed practice before the concept is understood.
2027 Singapore syllabus context
SEAB lists G2 Additional Mathematics as K232 and G3 Additional Mathematics as K341 for 2027. Check the student’s school-confirmed subject level and applicable official syllabus.
Frequently asked questions
Does one mistake mean the entire topic is weak? No. Identify the first incorrect step. Should parents immediately add tuition? Not automatically; seek targeted school clarification first. How can progress be checked? Solve changed questions independently after a delay.
Related reading
Related Additional Mathematics guide. SEAB G3 syllabus. SEAB G2 syllabus.
Arrange a consultation
Contact eduKate Singapore with the student’s original working to discuss suitable academic support.