Show WorkingWorked example 1: function notation
f(x) = 4x − 3
Work out f(−5).
- f(−5) means put x = −5 into the expression.
- f(−5) = 4(−5) − 3 = −23.
Answer −23
AlgebraHigher
Functions GCSE questions with answers
8 functions questions to try, each with the answer and the full working one tap away. Every question here was made and worked out by the Show Working maths engine, not typed in by hand.
Function questions on Higher use notation like f(x) = 3x + 1. You work out values such as f(4), find inverse functions and combine two functions into a composite function like gf(x).
01 / Quiz yourselfTry it, then tap to check
Functions questions to try
Work each one out on paper first. Then open the answer and compare your working with ours, step by step.
01 / Function notationHigher only
f(x) = x2 + 9
Work out f(3).
Show the answer and working
- f(3) means put x = 3 into the expression.
- f(3) = (3)2 + 9 = 18.
Answer 18
02 / Function notationHigher only
f(x) = 2x2 − 2
g(x) = −3x − 3
Work out gf(4).
Show the answer and working
- gf(4) means work out f(4) first, then put the answer into g.
- f(4) = 2(4)2 − 2 = 30.
- g(30) = −3(30) − 3 = −93.
Answer −93
03 / Function notationHigher only
f(x) = −3x + 2
Work out f(4).
Show the answer and working
- f(4) means put x = 4 into the expression.
- f(4) = −3(4) + 2 = −10.
Answer −10
04 / Function notationHigher only
f(x) = 6x − 4
Work out f(6).
Show the answer and working
- f(6) means put x = 6 into the expression.
- f(6) = 6(6) − 4 = 32.
Answer 32
05 / Inverse and composite functionsHigher only
f(x) = −2x − 8
g(x) = 5x + 6
Find fg(x). Simplify your answer.
Show the answer and working
- fg(x) means put g(x) into f: replace every x in f(x) with (5x + 6).
- fg(x) = −2(5x + 6) − 8.
- Expand and simplify: −10x − 20.
Answer fg(x) = −10x − 20
06 / Inverse and composite functionsHigher only
f(x) = x6 − 1
Find f−1(x).
Show the answer and working
- Write y = x6 − 1 and make x the subject.
- Add 1: y + 1 = x6. Multiply by 6: x = 6(y + 1).
- Swap y for x: f−1(x) = 6(x + 1).
- Check: put a number into f, then put the answer into the inverse. You get the number back.
Answer f−1(x) = 6(x + 1)
07 / Inverse and composite functionsHigher only
f(x) = 4x + 2
Find f−1(x).
Show the answer and working
- Write y = 4x + 2 and make x the subject.
- Subtract 2: y − 2 = 4x. Divide by 4: x = y − 24.
- Swap y for x: f−1(x) = (x − 2)÷4.
- Check: put a number into f, then put the answer into the inverse. You get the number back.
Answer f−1(x) = (x − 2)÷4
08 / Inverse and composite functionsHigher only
f(x) = x + 73
Find f−1(x).
Show the answer and working
- Write y = x + 73 and make x the subject.
- Multiply by 3: 3y = x + 7. Subtract 7: x = 3y − 7.
- Swap y for x: f−1(x) = 3x − 7.
- Check: put a number into f, then put the answer into the inverse. You get the number back.
Answer f−1(x) = 3x − 7
02 / What's covered2 kinds of question
Functions at GCSE and where marks go
- Function notationHigher only
- Inverse and composite functionsHigher only
Worked example 2: inverse and composite functions
f(x) = x2 + 9
Find f−1(x).
- Write y = x2 + 9 and make x the subject.
- Subtract 9: y − 9 = x2. Multiply by 2: x = 2(y − 9).
- Swap y for x: f−1(x) = 2(x − 9).
- Check: put a number into f, then put the answer into the inverse. You get the number back.
Answer f−1(x) = 2(x − 9)
A common mistake
Doing a composite function in the wrong order. gf(x) means do f first, then g. If f(x) = 3x + 1 and g(x) = 5x − 3, then gf(x) = 5(3x + 1) − 3 = 15x + 2.
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