Quick revision notes · Mathematics 0580 · Chapter 5 of 37
Functions
Function notation, inverses, composite functions, domain, range and asymptotes.
4 sectionsWorked examples
Common mistakesIGCSE: whole chapter is Extended tier
At a glance
Function notation, inverses, composite functions, domain, range and asymptotes.
1Function notation
- f(x) = 3x − 4 is a rule: f(2) means put 2 in place of x: f(2) = 2.
- Solve f(x) = 11 by writing 3x − 4 = 11.
- f(x + 1) means replace x by (x + 1): 3(x + 1) − 4 = 3x − 1.
- 2f(x) = 2(3x − 4) = 6x − 8 but f(2x) = 6x − 4.
Worked example
g(x) = x2 − 2x. Find g(−3) and solve g(x) = 8.
- g(−3) = 9 + 6 = 15.
- x2 − 2x − 8 = 0 → (x − 4)(x + 2) = 0.
g(−3) = 15; x = 4 or x = −2
2Inverse functions
- f⁻¹ undoes f: if f(3) = 11 then f⁻¹(11) = 3.
- f⁻¹(x) is not 1/f(x).
How to do it
- Write y = f(x).
- Swap x and y.
- Make y the subject.
- Write f⁻¹(x) = …
Worked example
f(x) = (2x − 5)/3. Find f⁻¹(x).
- y = (2x − 5)/3; swap: x = (2y − 5)/3.
- 3x = 2y − 5.
- 2y = 3x + 5.
f⁻¹(x) = (3x + 5)/2
Exam tip: Check: f⁻¹(f(x)) must give x back. Try a number.
3Composite functions
- fg(x) means f(g(x)): do g first, then f.
- Replace the x in f with the whole expression g(x), in brackets.
- ff(x) means apply f twice. fg and gf are usually different.
Worked example
f(x) = 2x + 1 and g(x) = x2. Find fg(x), gf(x) and fg(3).
- fg(x) = f(x2) = 2x2 + 1.
- gf(x) = g(2x + 1) = (2x + 1)2 = 4x2 + 4x + 1.
- fg(3) = 2 × 9 + 1 = 19.
fg(x) = 2x2 + 1, gf(x) = 4x2 + 4x + 1, fg(3) = 19
4Domain, range and asymptotes
- Domain: the x values allowed. Leave out any value that makes a denominator zero.
- Range: the y values you get out. x2 is never negative, so the range of f(x) = x2 is f(x) ⩾ 0.
- y = k/(x − a) + b has asymptotes x = a (vertical) and y = b (horizontal).
- y = ax + b has the horizontal asymptote y = b.
Worked example
f(x) = 3/(x − 2) + 1. State the value excluded from the domain and the asymptotes.
- x − 2 = 0 when x = 2.
- As x gets large, 3/(x − 2) → 0, so y → 1.
x = 2 excluded; asymptotes x = 2 and y = 1
✗Common mistakes
| Don’t write | Write instead |
|---|---|
| fg(2): working out f first | fg means f(g(2)): do g first |
| f⁻¹(x) = 1/f(x) | The inverse undoes f; it is not the reciprocal |
| gf(x) = 2x + 12 | Use brackets: (2x + 1)2 |
| 2f(x) and f(2x) treated as the same | 2f(x) doubles the output; f(2x) doubles the input |
★In the exam
- Use brackets every time you substitute an expression.
- Check an inverse with a number: f(2) then f⁻¹ of the answer should give 2.
- For 'solve ff(x) = …' find ff(x) as an expression first.