IGCSE Mathematics: Functions Practice Questions
Function notation f(x) means the rule f applied to x. A composite function fg(x) means apply g first and then f, and the inverse function reverses the original rule.
Topic 2.9 of Cambridge IGCSE Mathematics 0580 is an Extended topic where the notation itself is the difficulty. The order of operations in a composite function is the wrong way round from how it reads, and that catches almost everyone once. The questions below make the order explicit.
What you need to know for Functions and Proportion
- Function notationf(x) = 3x minus 2 means the rule multiplies the input by 3 and then subtracts 2. To find f(4), replace every x with 4.
- Composite functionfg(x) means apply g to x first, then apply f to the result. The function nearer to x is applied first.
- Order mattersfg(x) and gf(x) are usually different. Always check which function is written next to the bracket.
- Inverse functionf-1(x) reverses f. Write y = f(x), swap x and y, then rearrange to make y the subject.
- Checking an inverseApplying f then f-1 returns the original number. Test with any convenient value.
- Solving f(x) = kSet the expression equal to k and solve the resulting equation for x.
IGCSE Mathematics Functions and Proportion questions and answers
4 exam-style questions written to the 0580 syllabus. Try each one on paper first, then open the worked answer to check your method against the marks.
f(x) = 3x minus 2. Find f(4) and f(minus 1).
Show the worked answer
- To find f(4), replace x with 4: f(4) = 3 x 4 minus 2.
- 3 x 4 = 12, and 12 minus 2 = 10.
- To find f(minus 1), replace x with minus 1, keeping brackets: f(minus 1) = 3 x (minus 1) minus 2.
- 3 x (minus 1) = minus 3, and minus 3 minus 2 = minus 5.
f(x) = 3x minus 2. Find the inverse function f-1(x).
Show the worked answer
- Write the function as an equation: y = 3x minus 2.
- Swap x and y to reverse the roles of input and output: x = 3y minus 2.
- Rearrange to make y the subject. Add 2 to both sides: x + 2 = 3y.
- Divide by 3: y = (x + 2) divided by 3, so f-1(x) = (x + 2) divided by 3. Check: f(4) = 10, and f-1(10) = 12 divided by 3 = 4.
f(x) = 3x minus 2 and g(x) = x2. Find fg(3) and gf(3), and comment on your answers.
Show the worked answer
- For fg(3), apply g first because it is nearer to the bracket: g(3) = 32 = 9.
- Now apply f to that result: f(9) = 3 x 9 minus 2 = 25. So fg(3) = 25.
- For gf(3), apply f first: f(3) = 3 x 3 minus 2 = 7.
- Now apply g: g(7) = 72 = 49. So gf(3) = 49.
- The two answers are different, which shows that composite functions are not commutative. The order in which the functions are applied changes the result.
f(x) = 3x minus 2. Solve f(x) = 13.
Show the worked answer
- Replace f(x) with its rule and set it equal to 13: 3x minus 2 = 13.
- Add 2 to both sides: 3x = 15.
- Divide by 3: x = 5.
- Check by substituting back: f(5) = 3 x 5 minus 2 = 13. Note that this could also be found using the inverse: f-1(13) = 15 divided by 3 = 5.
Common mistakes in this topic
- Applying the functions in the wrong order in a composite.
- Reading f to the power minus 1 as a reciprocal.
- Treating f(4) as multiplication.
- Evaluating f at the value when the question asks you to solve for it.
- Losing brackets when substituting a negative input.
Exam tips
- In fg(x), the function next to x goes first. Read the notation from the inside out.
- For an inverse, write y equals, swap x and y, then make y the subject. Three fixed steps.
- Check an inverse by feeding a number through f and then through the inverse. You should get back where you started.
- Show the intermediate value in a composite. It is usually worth a method mark.
- Watch whether the question says find f(k) or solve f(x) = k. They are opposite operations.
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Functions and Proportion FAQs
What does fg(x) mean?
It means apply g first and then apply f to the result. The function written closest to the bracket acts first. This reads the opposite way round from how it appears, which is why fg(x) and gf(x) usually give different answers.
How do I find an inverse function?
Write the function as y equals the rule, swap x and y, then rearrange to make y the subject. The result is the inverse. Check it by putting a number through the original function and then through the inverse, which should return the starting value.
Does f to the power minus 1 mean one divided by f?
No. In function notation, f to the power minus 1 denotes the inverse function, the rule that undoes f. It is not the reciprocal. If f(x) equals 3x minus 2, then the inverse is (x + 2) divided by 3, not 1 divided by (3x minus 2).
What is the difference between finding f(5) and solving f(x) = 5?
Finding f(5) means substituting 5 as the input and calculating the output. Solving f(x) = 5 means finding which input produces an output of 5, which requires forming an equation and solving it. They are opposite operations.
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Written to the published Cambridge IGCSE Mathematics (0580) syllabus. Check your school entry code and syllabus year, because Core and Extended candidates are assessed on different content. Last reviewed 2026-08-12.